[v. Zubov, V Shalnov] Problems in Physics(BookZZ.org)

V. Zubov and V. Shalnov PROBLEMS IN PHYSICS

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B. l`. BYBOB u B. H. HIAJIBHOB 3A]]QA'~II/I HO CDI/IBI/IHE H0c06ue Bm; ca.M006pa306aH.u.a H30ame./zbcm60 < 10"’· cgs electrostatic units are put on mercury drops of radius r = 0.1 cm. Ten such drops merge into one large drop. What is the potential of the large drop? 351. Three capacitors of capacitance 0.002, 0.004 and 0,006 p.F are connected in series. Can a potential of 11,000 V be applied to this battery? What voltage will be received by each capacitor in the battery. The puncture voltage of each capacitor is 4,000 V. 352. Calculate the capacitance of a capacitor (see Problem 334) with a metal bar inserted into it if the area of each plate of this capacitor is S = 100 cm? and all the free space in the capacitor is filled with kerosene (s = 2.1). Will the capacitance of the capacitor be changed if the bar is moved parallel to itself from one plate to the other? 353. In a concentric-shape capaci— /___ , ·~ tor, the external and internal spheres are alternately connected to earth Fig_ 112 (Fig. 112). Will the capacitance of this capacitor be the same in these two cases? 354. The charges on each plate of a plane capacitor are acted upon by the electric field set up by the charges on the other plate. Theoretical calculations show that the intensity

92 1;>RoBLEMs of such a field set up by the charges of one plate in the capacitor is equal to E = 2n -gwhere Q is the charge on the plates and S is the area of each late. P If Q and S are known, determine the force with which the plates of the plane capacitor will be mutually attracted. What work is needed to draw the capacitor plates apart by a distance d. Express this work in terms of the capacitance of the capacitor and the potential difference and in terms of the capacitance and the charge on the plates. 355. Find the charge density on the plates of a plane capacitor if its capacitance is 100 cm, the distance between the plates 2 mm and if the plates are mutually attracted A by a force of 40 gf. Note. See Problem 354. 356. One of the plates of a plane capacitor is suspended 1 from the beam of a balance (Fig. 113). The distance between the capacitor plates is lid d = 5 mm and the area of the plates is 628 cm?. Fig 1.1.3 What is the potential dif` ference between the capacitor plates if a weight P = 0.04 gf has to be placed on the other pan of the balance to obtain equilibrium? Note. See Problem 354. 357. The plates of a plane capacitor are first drawn apart all the time being connected to a voltage source and then being disconnected after receiving the initial charge. In which of these two cases is more work required to draw the plates apart? Note. See Problem 354. 358. A plane air ·capacitor is charged to a certain potential difference. A dielectric bar is put into the capacitor. The charge on the bar must then be increased three times to restore the former potential difference. Determine the permittivity of the bar,

CHAPTER III. ELECTRICITY 93 23. The Laws of Direct Current This section requires special attention when solving problems involving the application of Ohm’s law to an electric circuit containing several sources of e.m.f., and also the problems on current intensities in branched circuits and in the sections of a circuit containing the sources of e.m.f. The errors in these problems usually occur because students confuse the signs ,of e.m.f. acting in a circuit and the directions of the currents and the electric field forces when writing the equations of Ohm’s law. Serious errors are also made when students forget that the intensity of current flowing in a section of a circuit with an acting source of e.m.f. is determined by the joint action of the electric potential difference at the ends of the section and the electromotive force of the source inside the section. For example, when calculating the current flowing through a storage battery of e.m.f. 8 and resistance R when it is charged from the mains having potential V, it is a common error to determine the current from the equation IR = V, frequently disregarding the proper equation IR = V — E. Errors are frequently caused by an inability to calculate the effect of the internal resistance of e.m.f. sources on the general functioning of the entire circuit. A number of problems in this section (for instance, 383, 385, 386, 392-395, etc.) are included to clarify this very point as well as the question of the best conditions for the operation of current sources. When solving most of the problems in this section one should watch how the distribution of currents and potentials changes when separate resistances or sources are introduced or replaced in the circuit. It will be difficult to solve such problems as, for example, 379 and 380 if you do not know the relationship between the distribution of potentials and currents in branched circuits. It is also important to show how and in what conditions one and the same measuring instrument can be utilized for various purposes (for example, the use of an ammeter as an ohmmeter, or a milliammeter as a voltmeter), and specify the most typical errors that can be made during measurement i¤ various conditions (Problem 375). For this reason, Some

94 PROBLEMS problems at the beginning of this section are based on the theory and operation of electrical measuring instruments. 359. The resistivity of copper is 1.7 >< 10‘° ohm·cm. What is the resistivity of a copper wire 1 m long and 1 mm? in cross section? 360. One of the first attempts to get a standard unit of measurement for the resistance of conductors in all laboratories was made by the Russian Academician B. S. Yakobi. Yakobi’s unit of resistance is equal to the resistance of a copper wire 6.358 feet long (1 ft = 30.5 cm) and 0.00336 in in diameter (1 in:-2.54 cm). Express Yakobi’s unit of resistance in ohms. 361. The resistance of a constantan wire is 10 ohms. Express this resistance in the units of the cgs electrostatic system. 362. How will the resistance of a telegraph line change from winter to summer if it is made of an iron wire 10 mm? in cross section? The temperature changes from —30°C to —l—30°C. In winter the length of the wire is 100 km, the resistivity of iron po = 8.7 >< 10‘6 ohm ·cm and the temperature coefficient of resistance 6 = 6 >< 10*3 deg*1. How will the result change if the elongation of the wire on heating is taken into account? The coefiicient of linear expansion of iron is oc : 12 >< 10"* deg‘1. 363. An electric lamp with a tungsten filament is rated at 220 V and consumes 40 W. Determine the length of the filament in this lamp if its diameter is 0.01 mm. When the lamp burns the absolute temperature of the filament is 2,700°. The resistivity of tungsten which at O°C is po == 5 >< 10** ohm ·cm increases in proportion to the absolute temperature of the filament. 364. Determine the intensity of the current flowing in the electric lamp in the previous problem immediately after it is switched on. How much bigger will this current be than the current when the lamp burns normally? 365. A plane capacitor having plates of area S separated by a distance d is first filled with a dielectric of permittivity e and then with an electrolyte of conductivity R. Find the ratio between the capacitance of the capacitor in the first case and its conductivity in the second.

cHA1>·rER III. ELECTRICITY 95 366. In his experiments on the thermal effect of current the Russian Academician H.F.E. Lenz* assumed as a unit of current the intensity of that current which would evolve 41.16 cm? of detonating gas in one hour at a pressure of 760 mm Hg and a temperature of 0°C when this current was passed through acidified water. Express Lenz’s unit of current in amperes. The density of oxygen at a pressure of 760 mm Hg is d = 0.00143 g/cm?. 367. What should the resistances of the sections of a rheos-tat R,. R2 and R3 be (Fig. 114) in order to change the current passing through an instru- I? ment of resistance R0 = 30 ohms Z by 1 A when the rheostat slide AQ is shifted from one contact to /1:, another. The circuit is powered by a source of 120 V ¤___._../ 368. It is required to measure the resistance of a circuit opera- V ting at 120 V. There is only one galvanometer with a re- /70 sponse of 10‘5 A per division. How should the galvanometer be cut in to operate as an ohm- Fig. 114 meter? What minimum resistance of the circuit can be measured with such a galvanometer if its full scale has 40 divisions? Construct the entire scale of such an ohmmeter in ohms per division. Disregard the internal resistance of the instrument. 369. A certain circuit with resistance R = 100 ohms is powered by a direct-current source. An ammeter with an internal resistance R0 : 1 ohm is cut into the circuit to measure the current. What was the current in the circuit before the ammeter was cut in if the ammeter shows 5 A? 370. What must the resistance of a shunt to a galvanometer be to reduce the response of the latter by 20 times? The internal resistance of the galvanometer is R0 = 950 ohms. * Heinrich Friedrich Emil Lenz (1804-1865) is famous for his law on the thermal effect of current and his rule applied to the phenomena of electromagnetic induction underlying today the theory of electric phenomena.

96 1>noB1.EMs 371. A milliammeter with a 20 mA scale is to be used as an ammeter to measure current up to 5 A. Calculate the resistance of the shunt if the internal resistance of the milliammeter is 8 ohms. 372. A sensitive milliammeter is utilized as a voltmeter. Determine the scale division of this instrument in volts if its internal resistance is 500 ohms and each division of the scale corresponds to 1 mA. 373. A voltmeter of internal resistance 400 ohms connected for measurement to a section of a circuit with a resistance of 20 ohms shows a reading of 100 V. R Fig. 115 What is the error in the readings of the voltmeter if the current in the circuit remains constant before branching? 374. The circuit shown in Fig. 115 is used to measure the resistance R. An ammeter shows a current of 2 A and a voltmeter a potential difference of 120 V. b0 00 3 0 R E R;) bf g' Fig. 116a Fig. 116b What is the magnitude of the resistance R if the internal resistance of the voltmeter is R0 = 3,000 ohms? How large will the error in measuring R be, if the resistance of the voltmeter is assumed to be infinitely large in calculations?

cHAr»·rEa 111. ELECTRICITY 97 375. A resistance R is calculated from the readings of a voltmeter and an ammeter connected as shown in Fig. 116a and 116b without any corrections being introduced for the internal resistances of the instruments. Find the error which will be committed in measuring a resistance R ·-= 1 ohm using these circuits if the internal resistance of the ammeter is Ra = 0.1 ohm and of the voltmeter Rv = 1,000 ohms. What will the error be in measuring a resistance R = 500 ohms? Which circuit should be used to measure low and high resistances? _ //0 1/_ 376. A certain circuit with aresistance R, = 10,000 ohms is powered from a potentio— 3000*9 meter of resistance R0 = A 0 = 3,000 ohms (Fig. 117).A 5 voltage V = 110V is supplied to the potentiometer. /0000.0 Determine the voltage V fed into the circuit when the slide Fi 1 17 is in the middle of the poten- g' tiometer. 377. A 60 W lamp burns in a room and an electric heating appliance of 240 W is switched on. The voltage in the mains is 120 V. The resistance of the wires connecting the electrical devices in the room with the mains is R0 = 6 ohms. By how much will the voltage supplied to the lamp be changed when the heating appliance is switched on? 378. A room is illuminated by n electric lamps each of B D which consumes a current I 0. The distance of the lead-in from the mains cable is l metres, the resistivity of the wires _ ir Determine the minimum A 0 permissible cross section of the Fig. 118 wires if the voltage loss in the line should not exceed V, volts. 379. Two conductors AB and CD are connected to the branches of an energized circuit (Fig. 118). The position of the points A, B, C and D is selected in a s-uch a way that no 7-1218

98 Pnonnnius current flows along these conductors. After that the two bridges are connected by a wire EK. Will current flow in this case in the wire EK, and in the conductors AB and CD? What will happen to the potentials A at the points A, B, C and D? What will be the potentials at the points E and K? 380. By mistake, a galvano— ,/· meter and a switch were cut · into the circuit of a bridge as shown in Fig. 119. How can the balance of the bridge be established by observIII I I III III I ing the readings of the galvanometer when the switch is 0 closed and opened? 381. In the system of electric Fig. 119 units derived by Academician Lenz, the unit of e.m.f. is that e.n1.f. which generates a current equal to Lenz’s unit when the resistance of the circuit is equal to one of Yakobi’s units (see Problems 360 and 366). Convert Lenz’s unit for e.m.f. to volts. 382. The e.m.f. of a storage battery is 6 V When the battery is connected to an external resistance of 1 ohm it produces a current of 3 A. What will the current be when the battery is shortcircuited? 383. An electric lamp of 110 V and 60 W is connected to a dry 120-volt storage battery. The internal resistance of the battery is 60 ohms. Will the lamp burn at full intensity with this kind of connection? 384. What is the internal resistance of a storage battery if it produces a current of 1 A when the resistance of the external circuit is 1 ohm and a current of 0.5 A when the resistance is 2.5 ohms? 385. In order to determine the e.m.f. of a storage battery it was connected in series with a standard cell to a certain circuit and a 0.2-A current I , was obtained. When the storage

CHAPTER III. nLEc·rR1c1TY 99 battery is connected to the same circuit Opposite to the Standard cell a 0.08-A current I 2 flowing in the External circuit from the positive pole of the Storage battery was obtained. What is the e.m.f. of the storage battery? The e.m.f. of the standard cell is $2 = 2 V 386. What should the e.m.f. of the storage battery in Problem 385 be to obtain a 0.08-A current flowing in the external circuit from the negative to the positive pole of the storage battery when it is out in the same direction as the standard cell? 387. The e.m.f. of a storage battery is 2 V, its internal resistance 0.4 ohm and the resistance of the external circuit 1 ohm. Determine the potential diffe- 8 rence across its terminals. ’ 388. A standard cell $1, a potentiometer witha resistance R = 10 ohms, a storage battery with an unknown 5 e.m.f. $2 and a galvanometer G are A connected as shown in Fig. 120. Indicate the position of the slide A on the potentiometer at which no cur- 8 rent will pass through the galvano- ’ meter. Determine the e.m.f. of the Fig- 120 storage battery if the current ceases to flow through the galvanometer when the resistance in the section of the potentiometer AB = 9 ohms. In this case the cell $1 produces a potential difference V0 = 2 V at the ends of the potentiometer. 389. In the circuit in the previous problem (Fig. 120) the potentiometer has a scale 50 cm long with millimetre divisions, the response of the galvanometer is 10*4 A per division, and the internal- resistance of the storage battery is r -:- 0.5 ohm. What should the resistance of the galvanometer be to detect the disturbance of equilibrium when the slide is shifted from the position of equilibrium by one division of the potentiometer scale? 390. With the external circuit cut in, the potential difference across the poles of a storage battery is equal to 9 V and the current in the circuit 1.5 A. What is the internal resistance r of the storage battery

100 PnoBLEMs and the resistance of the circuit R? The e.m.f. of the battery is 15 V. 391. Two identical storage batteries having e.m.f.s of 1.8 V and the same internal. resistances are connected as shown in Fig. 121. Determine the potential difference which will be established between points A and B. Disregard the resistance of the connecting wires. 392. A certain circuit with a resistance A B R is supplied with power simultaneously from N identical storage batteries. At what internal resistance of the storage batteries will the current in the cir_ cuit be the same when they are connected in F'g· 121 series and in parallel? 393. How many 100 V, 50 W electric lamps connected in parallel can burn at full intensity when supplied from a storage battery- with an e.m.f. 3 = 120 V and internal resistance r == 10 ohms? 394. How many storage batteries of e.m.f. 2 V and internal resistance 0.2 ohm should be connected in series to obtain a current I = 5 A in the external circuit with a potential difference of V = 110 V across the poles of the battery? 395. When the resistance of the external circuit is 1.0 ohm the potential difference across the terminals of a storage battery is 1.5 V When the resistance is 2 ohms the potential difference increases to 2 V Determine the e.m.f. and the internal resistance of the storage battery. 396. A storage battery of e.m.f. 6 V and internal resistance r = 1.4 ohms supplies power to an external circuit consisting of two parallel resistances of 2 and 8 ohms. Determine the potential difference across the terminals of the storage battery and the currents in the resistances. 397. An external circuit with a resistance 0.3 ohm is powered by six storage batteries each of.e.m.f. 2 V and internal resistance 0.2 ohm. The storage batteries are connected as separate groups in series and the groups are then connected in parallel. What method of connecting the storage batteries in such

CHAPTER III. Emzcrnrcrrv 101 groups will provide the highest current in the circuit? What will this maximum current he? 398. A circuit with an external resistance R is powered by a storage battery consisting of N cells. The e.m.f. of each cell is $0 and the internal resistance is ro. The storage battepy is composed of identical groups connected in series. In turn, the groups consist of elements connected in parallel. Find the number of the groups n and the number of the cells m in each group for which the maximum current will be observed in the circuit. 399. A current of 8 A is required in a circuit with a resistance of 5 ohms. What is the minimum number of storage batteries needed to provide this current and how should they be connected into a compound battery if the e.m.f. of each battery is 2 V and the internal resistance 0.5 ohm? 400. A storage battery is connected to the mains for charging with voltage of 12.5 V (Fig. 122). The internal resistance of the storage battery is 1 ohm. /25 V 8: ii] Fig. 122 Fig. 123 What is the e.m.f. of this storage battery if a current of 0.5 A flows through it during charging? 401. A storage battery discharged to 12 V is connected for charging to 15—volt mains. What auxiliary resistance should be connected to the circuit so that the charging current does not exceed 1 A? The internal resistance of the storage battery is 2_ ohms. 402. A dynamo with an e.m.f. E1 = 120 V and an internal resistance r = 0.5 ohm, and a storage battery with an e.m.f. gz = 110 V are connected to an external resistance R as shown in Fig. 123.

102 pnonmms At what maximum value of R will no current pass through the storage battery? How will the battery operate when the resistance R is larger or smaller than the calculated value? 403. The e.m.f. of a storage battery is $1 = 90 V before charging and $2 = 100 V after charging. When charging began the current was 10 A. What is the current at the end of charging if the internal resistance of the storage battery during the whole process of charging may be taken as constant and equal to 2 ohms and the voltage supplied by the charging plant as directcurrent voltage? 24. Thermal Effect of Current. Power Even if the reader is well versed in the Joule-Lenz law, he may sometimes fail to find the proper form of writing this law when solving some problems. For example, when solving Problem 411 on heating water in an electric kettle heater with two coils, students frequently forget that the voltage at the ends of the coils remains constant whatever their method of connection and they attempt to obtain the required results by applying the formula Q = 0.24 I2Rt instead of the more convenient formula Q = 0.24 Egt. In analysing the solutions to the problems in this section it is worth considering the features of each electric circuit to select the most convenient form for writing the J oule—Lenz law, noting the difference in the physical meaning of the forms of writing this law. As we know the work done by the forces of an electric field in a given section of a circuit when a current is passed, is determined by the ratio A = I Vt while the amount of heat liberated in this section can be found from the equation Q = I2Rt. If the section of the circuit being considered does not contain any sources of e.m.f. all the work done by the electric field forces is wholly expended on liberating Joule heat and both ratios produce the same results. If a source of e.m.f. is present inside this section some of the work done by the electric field forces is expended to overcome these electromotive forces and the relationships mentioned above yield different results.

cnnprnn III. Enncrnrcirv 103 Since the work done by the current in a section of a cir·cuit having a source of e.m.f. is not included in the school curriculum the difference in the physical meaning of the formulas A = I Vt and Q = I 2Rt usually escapes the attention of the student and he encounters serious difficulties in solving problems such as 409 and 410. Before solving problems of this kind refresh your knowledge of the work done by electric field forces and the thermal action of current. A number of problems in this section (for instance, 406408, 416, 417, etc.) are intended to draw the attention of the student to the nature of the dependence of the efficiency and useful power of current sources on the ratio between the resistance of an external circuit and the internal resistance of a source. This dependence is not always clear to some pupils who thus fail to give exhaustive answers to questions requiring calculation of the most favourable conditions for operation of current sources. 404. In one of his experiments on the thermal effect of current Academician Lenz heated 118 g of the alcohol with a current of 15.35 Lenz’s units (see Problem 366). Determine how long it took to raise the temperature of the alcohol by 1° if the resistance of the coil is 35.2 Yakobi’s units (see Problem 360). The specific heat of the alcohol is 0.58 cal/g·deg. Disregard heat losses. 405. Use the data from the previous problem to calculate the time needed to heat the alcohol by 1°C with a current of 1 A and with a coil resistance of 1 ohm. 406. The coil of a heater has a resistance of 5 ohms and is supplied from a current source of internal resistance 20 ohms. What should be the resistance of the shunt in the heater to reduce the amount of heat liberated in the heater to one ninth of the value without a shunt? 407. A storage battery of e.m.f. 12 volts and internal resistance r = 0.8 ohm supplies in turn external circuits rated at 0.4, 0.8 and 2 ohms. Calculate for each of these three cases the useful power supplied by the battery and its efficiency. Explain the

104 raonmms nature of the dependence of the efficiency and useful power on the resistance of the external circuit. 408. At the end of his article "Liberation of Heat in Conductors" Academician Lenz offers the following problem: "A circuit consisting of n cells is required to heat a wire of a definite diameter and length Z. How many of these cells are needed to heat a wire of the same diameter but with a length pl?" In both cases the elements are connected in series. Solve this problem. 409. The voltage in the mains of a charging plant is 13 V. The internal resistance of a storage battery being charged is 0.4 ohm and its residual e.m.f. is 11 V. What power will be expended by the plant in charging; this storage battery? What part of this power will go on heating the storage battery? 410. An electric motor has an ohmic resistance of 2 ohms and is driven from mains of 110 V. When the motor is running the current passing through it is 10 A. What power is consumed by this motor? What fraction of this power is converted into mechanical energy? 411. An electric kettle heater has two coils. When one coil is switched on, the water in the kettle begins to boil after 15 minutes and when the other is switched on—after 30 minutes. How soon will the water in the kettle begin to boil if both coils are connected: (a) in series; (b) in parallel? 412. A current is passed along an iron wire with such an intensity as to heat it noticeably. Explain why the cooling of one part of the wire (with water, for example) causes the other part to become more intensely heated than before the first part was cooled. The potential difference at the ends of the wire is kept constant throughout. 413. A fuse made of a lead wire with a cross section of 0.2 mm? is incorporated into a circuit of copper wire with a cross section of 2 mm2. On short—circuiting the current reaches 30 A. How long after the short-circuit occurs will the lead fuse begin to melt? How much will the copper wires heat up during this time? Neglect the loss of heat due to thermal conductivity. Take the specific heat of lead as constant

CHAPTER III. nnncrnrcrrv 105 and equal to C1 = 0.032 cal/g·deg and of copper, C2 :.=-0.091 cal/g·deg. The resistivity of lead is pi = 22 >< >< 10*6 ohm·cm. The melting point of lead is T , = 327°C. The temperature of the wires before short-circuiting is T0 = 20°C. The density of copper is dz = 8.9 g/cm3 and of lead di = 11.34 g/cm3, 414. In one calorimeter there is a certain amount of water and in another the same mass of a liquid whose heat capacity is to be determined. Identical constantan wires connected in series to a current-carrying circuit are immersed in the calorimeters. What is the specific heat of the liquid if the temperature of the water rises by 2.50°C and that of the liquid by 4.25°C some time after the wires are energized? 415. A steel wire has a resistance twice as great as a copper one. Which wire will liberate more heat: (a) in parallel; (b) in series connection with both wires in a circuit powered by a direct—current voltage? 416. A wire with a resistance R = 2 ohms is first connected to a storage battery of internal resistance ro = 2 ohms and then another such wire is connected in parallel. By how much will the quantity of heat liberated in the first wire change after the second is cut in? 417. A storage battery is shorted by an external circuit first with a resistance Ri and then with a resistance R2. At what value R0 of the internal resistance of the storage battery will the quantities of heat liberated in the external circuit be the same in both cases? 25. Permanent Magnets 418. Two magnetic poles repel each other with a force of 8 gf. The distance between the poles is 10 cm. The magnetic mass of one pole is twice as large as that of the other. Determine the magnitude of the magnetic masses of these poles. 419. A bar magnet has a length Z = 10 cm and the magnetic masses of its poles are m = 10 cgs electromagnetic units.

106 pnonrnms Determine the magnitude and direction of the field intensity vector produced by such a magnet at point A lying on the extension of the magnet’s axis at a distance 0. = 5 cm from the south pole. 420. Two identical magnets with a length Z : 5 cm and weight p = 50 gf each made of magnico alloy which was obtained by the Soviet scientists Zaimovsky and Lvov are arranged freely with their like poles facing in a vertical glass tube (Fig. 124). The upper magnet hangs in the air above the lower one so that the distance a between the nearest poles of the magnets is 3 mm. S ;L?—i .L A .._ ‘_‘— Fig. 124 Fig. 125 Determine the magnetic masses of the poles of these magnets. Will remote poles change the distance between the magnets?‘ 421. In order to keep the needle (Fig. 125) i·n a horizontal position a load of 0.01 gf is suspended from its top end. Find the magnitudes of the horizontal and vertical components of the intensity of the terrestrial magnetic field. Calculate the total intensity of the terrestrial magnetic field. The dip angle oe = 70°. The magnetic masses of the poles of the needle are m == 9.8 cgs electromagnetic units. 422. If a magnetic needle is secured to a cork and the cork is floated on water the terrestrial magnetic iield will cause the needle to turn and set itself along the magnetic meridian, but the needle will not move northwards or southwards. lf the pole of a bar magnet is placed not far from the needle the field of the magnet will set the needle

in the direction of the lines of force and will cause it to move towards the magnet. What are the reasons for the different behaviour of the needle in the magnetic fields of the earth and the magnet? 423. One is given two outwardly identical long barsone is made of soft iron and the other is a steel magnet. By observing the interaction of the bars in various positions how can one determine which of them is a magnet. 424. The length of a thin bar magnet is Z = 10 cm and the magnetic ¢ masses of the poles are m = i 50 cgs . electromagnetic units. __ Determine the force acting on a ` J unit north magnetic pole at a point gg A lying on the perpendicular to the axis of the magnet in its middle point. · The point A is at a distance 10 cm CI L from the axis. Consider the poles as points. WSI 425. A dip needle with a circular .,·é:g scale is secured on a horizontal axis (see Fig. 125). lm" N N How can the direction of the mag- . . netic meridian be found with the aid Flg‘ 126 F1g° 127 of this needle? 426. Several steel needles are freely suspended on hooks from a small brass disk as shown in Fig. 126. If the pole of a strong magnet is brought up to the needles from below the needles will first be drawn apart and then will again assume a vertical position when the magnet is brought right up to them. As the magnet is removed, the needles will again be drawn apart forming a cone—shaped bunch. Explain the causes of this behaviour. 427. Two long equally magnetized needles are freely suspended by their like poles from a hook as shown in Fig. 127. The length of each needle is 20 cm and the weight 10 gf. In equilibrium the needles make an angle on == 2° with each other. Determine the magnetic masses of the poles of the needles. The magnetic masses are concentrated at the ends of the needles.

mg pRoBLEMs 428. The following experiment can be carried out with the aid of strong magnets made of magnico alloy. Identical magnets A and B are placed along one straight line with their like poles fitted tightly against each other. Then magnet B is raised so that it rests on its edge. The magnet will be held in equilibrum in this tilted position by the forces of interaction of the poles NN (Fig. 128). t A Fig. 128 Determine the force of interaction and the magnetic masses of the poles of the magnets when the magnet B of length l = 10 cm and weight P = 100 gf is in equilibrium at an angle oc = 10° The magnetic masses of the poles can be taken as points and as being situated at the ends of the magnets. Will the position of equilibrium of the magnet B be stable in this case? '""` 429. The magnetic moment M J P of a magnetic needle is the product of the needle lengthl . __’" and the magnetic mass m of -__Q ..._._.._—., one of the poles: P =· ml. Fig_ 129 The needle is placed in a uniform magnetic field of intensity H. The direction of the needle forms an angle cx with the direction of the lines of force of the field. Determine the magnitude of the mechanical moment acting on the needle in this case. Express the mechanical moment in terms of the magnetic moment of the needle and the intensity of the field. 430. A magnetic needle uniform along its length has a magnetic moment P = 50 cgs electromagnetic units and a weight Q = 5 gf. How should the point of support be arranged with resPect to the centre of gravity of the needle to set it in a hori-

cnxprnn III. ELECTRICITY 109 zontal position in the terrestrial magnetic iield in the northern hemisphere? The vertical component of the magnetic field is HU = 0.5 cgs electromagnetic units. 431. A magnetic needle with length Z and poles of magnetic masses im is attached to a wooden bar of length L (Fig. 129) and placed in a uniform magnetic field of intensity H. The bar and the needle can revolve around point O. Find the magnitude of the mechanical moment that will cause the rotation of the bar around point O if the bar makes an angle oc with the direction of the lines of force of the magnetic field. 432. A magnetic needle has a length l and the magnetic masses of its poles are im. The needle is broken in two. What will the magnetic moments of the ( two halves be? Mh;];·i.( 433. Draw the lines of force of the H1}glll|·| ‘ magnetic field inside a magnetized steel tube. al, 434. Indicate the positions and the will . . .( [ull. nature of equilibrium of a number of HM magnetic needles arranged in a straight line at equal distances from one an- (H; _A" other. |\i g 435. A strong horseshoe magnet is closed by an iron plate A (Fig. 130). ·—·—*· The weight of the plate corresponds to l the lifting force of the magnet, and the Fig_ 130 magnet can easily hold the plate. If the poles of the magnet are now touched on the sides with a plate B made of soft iron, the plate A will drop at once. If the plate B is removed the magnet will again be capable of holding the plate A. Explain this phenomenon. 436. A long rod made of soft iron is secured in a vertical position. If a strong magnet A is brought to the top of the rod as shown in Fig. 131 the rod will be magnetized so intensely as to retain at its other end several small pieces of iron. If the same magnet A is applied to the side. of the rod near the bottom end (Fig. 132) the magnetization will 'be weak and the pieces of iron will fall. Explain why the magnet A acts differently in these two cases.

437. A strong magnet of magnico alloy can hold a chain consisting of several cylinders made of soft iron (Fig. 133). What will happen to the cylinders if a similar magnet is brought up from below to this chain? The magnets are arranged with their like poles facing. What will happen to .l lll l }* ` Il I n, #/7 i `T i `“ 1 . 1 rl .饓.'ll ll l ll 5, l1l E `. .*‘l A l ll `I l llillllllllll ill 5 l wl l 1 ll T , ao i , =i *6 if r l M /{f/l p_ l $4, _ Fig. 131 Fig. 132 Fig. 133 Fig. 134 the cylinders if the magnets have their opposite poles facing? 438. Two identical horseshoe magnets are linked by their opposite poles as shown in Fig. 134. One of the magnets has round it a coil A whose ends are connected to a galvanometer G. If the magnets are detached, the pointer of the galvanometer will be deflected at this instant through a certain angle. If the magnets are connected again the pointer of the galvanometer will also be deflected, but this time in the opposite direction. Indicate the causes for the deflection of the pointer of the galvanometer. 439. Permalloy can be magnetized appreciably in the terrestrial magnetic field and does not possess residual magnetism, i.e., it is the softest material as far as magnetism is concerned.

CHAPTER III. ELECTRICITY Mi How will a magnetic needle on a vertical axis near a long bar made of this alloy behave if: (a) the bar is vertical (Fig. 135); (b) the bar is horizontally placed along the magnetic meridian; V lx wl if ii'., — Fig. 135 Fig. 136 (c) the bar is in a horizontal plane perpendicular to the magnetic meridian. Will the behaviour of the needle change in these three cases when the bar is turned? 440. A small thin iron nail is suspended from a light fire—proof thread. A strong electromagnet is placed near the nail (Fig. 136). The flame E from a powerful gas burner is U _ directed precisely between the lf: , nail and the magnet and licks fil E" i the nail when it is deflected by e i *2; the magnet. If the windings of J the electromagnet are energized " sb; lZ·l1€ Hall be at OHC9 (l€l'l€Cl.9d into the flame and will then be ejected from it to assume its Fig.137 original position. After a lapse of time the nail will again be drawn to the magnet. Explain what causes these periodic oscillations of the nail.

Mg PnoBLEMs 441. F.N. Shvedov suggested the following design for a motor. Some 20-30 nickel rods similar to these used in an umbrella are attached to a small support bush htted onto a sharp point. Nearby are arranged a strong electromagnet and a gas burner with a broad and intensive flame as shown in Fig. 137. When the windings of the electromagnet are energized and the burner is ignited the impeller begins to rotate uniformly in the direction shown by the arrow on the drawing. Explain what makes the impeller rotate. 26. Magnetic Field of a Current Most of the problems in this section deal with the properties and peculiarities of the magnetic held of current related to the closing of the lines of force of this held. Since the problems are partly based on material outside the scope of the school curriculum these problems are recommended for discussion in extracurricular circles. Pay special attention to the problems on the work done by the forces of the magnetic held of current in a closed loop. 442. Draw the lines of force of a magnetic held of a rectilinear current. 443. A current I flowing along a sufhciently long rectilinear conductor sets up, as we know, a magnetic held with an intensity H :0.2% cgs electromagnetic units where r is the distance of the point in the field from the current—carrying conductor as measured in cm and I is the current in amperes. Determine the intensity of the held at point A 5 cm away from the conductor if the current is 2 A. Draw the field intensity vector. Find the force acting at point A on a magnetic pole of m. = 5 cgs electromagnetic units. 444. Given: (a) The intensity of the magnetic held of a rectilinear current at a distance R0 == 1 cm from the conductor is H0 Z 0.21..

cnaprnn III. Enncrnrcrrr M3 (b) The lines of force of the magnetic held of such a current are concentric circles. (c) When a unit pole moves along a closed circuit the work done by the magnetic held forces is zero if this circuit is not penetrated by currents. Use these data to derive the formula for the dependence of the intensity H of the magnetic held of the current on the distance R from the conductor. 445. A magnetic held at a certain point A is composed of the terrestrial magnetic held with a horizontal component of intensity H h = 0.2 cgs electromagnetic units and the magnetic held of rectilinear current I = 5 A. How should a current-carrying conductor be arranged with respect to the point A for the vector of the intensity of the resultant held be vertical at this point? 446. A conductor carrying a current is placed as in the previous problem. At what points in space will the held intensity be equal to zero if the vertical component of the terrestrial magnetic held is H ,, = 0.5 cgs electromagnetic units? 447. A current I flows along an inhnite T rectilinear thin-walled pipe. _ What is the intensity of the magnetic V held inside the pipe? The current is distributed uniformly along the entire section of the 1 \ gl \ 1>i1>¤· li 1 448. A current I flows upwards along the Ii inner conductor of a coaxial cable (Fig. 138) ' L and returns down along the external shell P-ig_ 138 of the cable. What is the intensity of the magnetic held at points inside the cable? 449. A magnetic pole of m. = 5 cgs electromagnetic units is passed around a circumference of radius R. A rectilinear conductor carrying a current of 2 A is laid perpendicularly to the plane of this circle through its centre. Calculate the work done by the forces of the magnetic held of the current during this displacement of the. magnetic pole. 450. When a magnetic pole moves along a closed path under the same conditions as the previous problem some 8-121:;

444 Pnonmms work is obtained. Can this result be used to construct a perpetuum mobile? 451. A circular branching made from a uniform conductor is placed in a d-c circuit (Fig. 139). What force will the magnetic field of the currents in the branching exert on the magnetic pole placed at its centre? 452. A cork floats in a broad vessel filled with a weak solution of sulphuric acid. Two small plates—copper and idly - ` ‘ . ( ( iis; . A12 tc ~ I Fig. 139 Fig. 140 zinc—are passed through the cork. The plates are connected on the top by a copper wire (Fig. 140). What will happen to the cork if the end of a strong bar magnet is brought up to it? 453. As we know, a current I flowing along a circumference of radius R creates in its centre a magnetic field of intensity 0.2:tI H; T Determine the force acting on a unit magnetic pole placed at the centre of a circular current of 5 A if the radius of the circle is R = 10 cm. Indicate the direction of this force, assuming the direction of the current to be known. 454. Professor A. A. Eichenwald of the Moscow University carried out one of the basic experiments which directly revealed the generation of magnetic fields for any displacement of electric charges. In his experiment a certain charge was put on a massive disk which was then set in rapid rotation. The magnetic field set up by the charge on the disk was detected with the aid of a magnetic needle arranged above the instrument (Fig. 141).

p CHAPTER III. Enmcrnrcrry M5 Determine the direction in which the needle is deflected if a negative charge. is put on the disk and the disk rotates in the direction shown. 455. Currents are sometimes measured with the aid of the so-called tangent galvanometer (Fig. 142) consisting of a small magnetic needle suspended from a light thread and placed in the centre of a circular current. The plane of the energized circle is arranged strictly in the plane of the magnetic meridian. /\ CX.`.`;..;`.`;.O Q Vi if/z Fig. 141 Fig. 142 Determine the angle through which the needle of the tangent galvanometer will turn if a current I = 1 A is passed along the circle, if the radius of the circle is R =-·· = 10 cm and the horizontal component of the terrestrial magnetic field is H h = 0.2 cgs electromagnetic units. Note. See Problem 453. 456. A current I is passed along the ring of the tangent galvanometer in Problem 455 so that it establishes a field H c = 0.1 cgs electromagnetic units in the centre of the ring. As the current is being passed the circle turns after the needle. Determine the angle through which the circle should be turned so that the needle is in the plane of this circle in the case of equilibrium. 457. The magnetic field intensity inside a solenoid is proportional to the current I and the number of turns per 8*

116 PROBLEMS unit length of the solenoid,. i.e., H: 1.261-il where N is the total number of turns of the solenoid and l is its length. Find the magnetic field intensity inside a long solenoid made of wire with a diameter 0.5 mm. The wire of the solenoid is wound so that the turns are close to one another. The current is 2 A. 27. Forces Acting in a Magnetic Field on Current·Carrying Conductors Even if the student uses correctly the left-hand rule when the direction of current is perpendicular to the lines of force of a magnetic field he usually encounters some difficulties in applying it for finding the direction of the force acting on a conductor when the current and the lines of force of a magnetic field form acute angles with each other. It is far more difficult to determine the nature of motion of a conductor when the lines of force form various angles with the direction of current in various sections of a conductor. It is just as difficult to take account of the effect exerted by the heterogeneity of a fneld on the behaviour of a conductor in the simplest cases. All the problems in this section deal precisely with these cases. In solving the problems, pay careful attention to the sequence of applying the left-hand rule when N 7 ppyp. 5 finding the forces acting on the separate elements of conductors in various conditions. 458. A rectilinear current·carr in conductor is arranged above the poles osf aiorseFig M3 shoe magnet as Shown in Fig. 143. The ° conductor can move freely in all directions. What will happen to the conductor under the action of the field of a magnet if the current passes in the direction indicated by the arrow?

onarrnn 111. nnncrnrcrrx M7 459. A flexible free conductor is placed near a strong long bar magnet (Fig. 144). How will the conductor arrange itself if current is passed through it from the top to the bottom? 460. Currents are passed through two free rectilinear conductors arranged at right angles as shown in Fig. 145. How will the interaction of the magnetic holds of the currents change the position of the conductors relative to each other? 461. A rectilinear current I 2 is passed along the axis of a circular current I , (Fig. 146). With what force are the currents interacting? 462. A soft spiral spring hangs freely. The lower end of the spring is immersed in a cup of mercury. The spring and the cup are connected to a d—c source as shown in Fig. 147. What will happen to the spring after the circuit is closed by a switch S? 463. A beam of positively charged particles moves with a velocity v into a uniform magnetic field perpendicularly to the lines of force of this field (only one particle is shown in Fig. 148). Along what path will the particles move in such a magnetic field? 464. An infinite rectilinear energized conductor AB has near it a movable uniform rectilinear conductor CD of finite length the whole of which lies on one side of AB and in a plane passing through AB (Fig. 149). What will happen to the conductor CD if current is passed through it in the direction indicated by the arrow? 465. Current is passed along the conductor CD from point D to point C (see the previous problem). How will the conductor CD move in this case? 466. Two vertical circular conductors with approximately equal diameters are arranged in mutually perpendicular planes as shown in Fig. 150. How will the conductors behave if a current is passed through them in the directions indicated by the arrows? 467. An energized wire ring is freely suspended from soft infeed conductors as shown in Fig. 151. A horizontal magnet is brought close to the ring. What will happen to the ring in this case?

F4 V. i 4/ 1 ··‘*·“‘· Ml.5 1W5 “ ji “" Fig. 144 Fig. 145 Fig. 146 S ., •U• • • ()————>— +6 O • • • Fig. 147 Fig. 148 iI 6 ie Ai Fig. 149 Fig. 150

CHAPTER III. ELECTRICITY 119 468. An energized ring (see the previous problem) is arranged in the middle of a magnet. What will happen to the ring if the direction of current is reversed in it? i • ·. A ‘ i .v.. { A { F E ' ( I¥¤3i'j;_(,,,.·» llilv gp, la. A — ( *\\·j= V) ·· . I I FFJHWEUIIHWII Fig. 151 Fig. 152 469. A copper disk is secured on a horizontal axis and placed between the poles of a strong magnet so that the north pole of the magnet is arranged on the right (Fig. 152). The bottom of the disk is immersed in a cup of mercury. Q xw N Avg: D' Fig. 153 Fig. 154 The axis of the disk and the cup are connected to a d—c source. What will happen to the disk when the circuit is closed? 470. A light rectangular frame is suspended from a thread near an infinite rectilinear conductor with current passing through it (Fig. 153).

120 pnonwms How will the frame behave if current is passed through it in the direction indicated by the arrows? 471. A rectangular frame with current passing along it is arranged in a uniform magnetic field so that its axis is perpendicular to the lines of force of the magnetic field (Fig. 154). Indicate the direction of the forces exerted on the sides of the frame BC and DA. Show how the magnitude of these forces changes as the position of the frame changes during rotation. 28. Electromagnetic Induction 4772. In his work "How To Determine the Direction of Induced Currents" in which Lenz’s famous rule was laid down for the first time, Academician H.F.E. Lenz describes some of his experiments which he carried out to determine E i )_/ A c .T;.· 8 l Fig. 155 Fig. 156 the direction of induced currents. In particular, he considers the case of an induced current generated in a circular conductor when it is turned through 90° relative to another circular conductor with current passing along it (Fig. 155). Determine the direction of current in a movable conductor A 1f .1t IS transferred from a position perpendicular to the circuit B to one parallel to it as indicated by the arrow. _ 473. A rectangular conductor AC of finite length is perpendicular to an infinite rectilinear current B (Fig. 156). The conductor AC Omoves along metal guidelines parallel to itself in the direction of current B.

CHAPTER III. ELECTRICITY 121 Indicate the direction of currents induced in the conductor AC when its direction of motion coincides with that of the current B. How will the induced current be directed if the conductor AC moves in the opposite direction? 474. Determine the directions of induced currents in the following experiment performed by Lenz. A permanent magnet is placed along the magnetic meridian. A rectilinear conductor is arranged parallel to the magnet first above it and then under it. The magnet is -—· — -=~ * ) ·—· ii · l"- " 'B ‘:'-\ '> / i .9 ( N * C -—-—— 0 V Fig. -157 Fig. 158 rapidly turned through 90° first with its north pole to the east and then to the west. 475. A copper disk is placed between the poles of magnets as shown in Fig. 152. A galvanometer is connected instead of a storage battery to the electric circuit shown in the diagram. In what direction will the induced current flow when the disk rotates: (1) clockwise; (2) counter-clockwise? 476. Two rectilinear parallel conductors are moved towards each other. A current I flows through one of them. What is the direction of the current induced in the other conductor? What is the direction of induced current when the conductors are drawn apart? 477. The south pole of a magnet is removed with a certain velocity from a metal ring as shown in Fig. 157. Determine the direction of the induced currents in the ring. 478. A small rectangular wire frame falls freely in the space between the wide poles of a sufficiently strong electromagnet (Fig. 158) "

122 PROBLEMS Show the direction of the currents induced in the frame when the middle of the frame passes through the positions A, B and C. How will the frame move in these sections? 479. A small pendulum consisting of a metal thread, a ball and a sharp point immersed in a cup of mercury ,., (Fig. 159) makes part of an electric circuit. The 0 — pendulum is placed in · _ g` the space between the ( _ ‘ Q t p_ broad poles of an elect‘ - romagnet and swings in _- _ the plane perpendicular S U N to the lines of force of the magnetic field. Du// / ring the oscillations the ’ -9* sharp point of the penFig. 159 dulum remains immersed in mercury. How will the magnetic field effect the motion of the pendulum? What is the direction of the currents in the circuit of the pendulum? T /->l§_ .·-·""'1_·_‘:;> `·—- -·"'% 1l \\ ai ....®,._ _ ·-— { e · »·* *·~-* ~---4-33 Fig. 160 Fig. 161 480. A copper wire connected to a closed circuit is surrounded by a thick iron shell (Fig. 160) and introduced together with the shell into the space between the poles of an electromagnet. The iron shell acts as a magnetic screen for the wire. Will an e.m.f. bc induced in the wire?

CHAPTER III. ELECTRICITY 123 481. An aircraft flies along the meridian. Will the potentials of the ends of its wings be the same? Will the potential difference change if the aircraft flies in any other direction with the same velocity? 482. A rectangular wire frame rotates with aconstant velocity around W, .....\ one of its sides parallel to a currentcarrying rectilinear conductor nearby y '``. , u___ (Fig. 161). §@1mmm1m1Hu1l|¤l1||ua11m1·:i ‘‘‘’·· Indicate the positions in which the * maximum and the minimum e.m.f.S will be induced in the frame. \\ _ ___.. . .·~~ 483. Two circular conductors are ```` perpendicular to each other as shown F1g_ 162 in Fig. 162. Will a current be induced in the conductor A if the current is changed in the circuit B?

Chapter IV OPTICS 29. The Nature of Light This section includes a number of problems which require the application of the simplest relations of wave and quantum optics. In solving these problems, attention should also be paid to the change in individual magnitudes (wavelength and velocity) characterizing a light wave moving from one medium to another. It is difficult to understand the ratio c = M if the nature of these changes and their physical meaning are not taken into account. Before solving these problems it is recommended to refresh your knowledge of the subject from a text-book. 484. The velocity of light c and the length of a light wave A are related to the frequency of oscillations v by the ratio c = PW Determine the change in the wavelength of red rays during the passage from vacuum to glass if the refractive index of glass is rz = 1.5 and the frequency of the red rays is v = 4 >< 101* s‘1. 485. The refractive index of any substance is equal to the ratio of the velocity of light propagation in vacuum to the velocity of light propagation in the given medium. It was found that the refractive index of one type of glass is equal to ni = 1.50 for red rays and to nz = 1.54 for violet rays. Determine the velocities of propagation of these rays in glass. 486. P. A. Cerenkov has found experimentally that when an electron moves in some medium with a constant velocity exceeding the velocity of light propagation in this medium it will begin to emit light.

cHAr>·rER iv. o1>·r1cs 12% Determine the minimum velocity to which the electron Should be accelerated to produce such an emission when the electron moves in a medium with a refractive index n, : $1.•5• 487. Explain the phenomenon of the coloured bands seen on thin films of oil on the surface of water. 488. If a thin soap film is arranged vertically the coloured horizontal interference bands move downwards and at the same time change their width. After some time a rapidly growing dark spot appears at the top of the film which bursts shortly afterwards. Indicate what causes the motion of the bands and explain the origin of the dark spot. 489. How will the pattern of Newton’s rings change if the space between a lens and a plane glass is filled with liquid whose refractive index is higher than that of the lens, but less than that of the glass? 490. "Antireflection Optics" which was developed by Academicians I.V. Grebenshchikov, A.A. Lebedev and A.N. Terenin is widely employed in present-day optical instruments to reduce the loss of light due to reflection from the surfaces of lenses. This method is based on the following phenomenon: if the surface of glass is coated with a thin transparent film whose refractive index is less than that of the glass and whose thickness is equal to a quarter of a wavelength of the incident light, the intensity of light reflected from such a plate will be zero and all of the light will pass through the plate. Consider the interaction of the light beams reflected from the upper and lower surface of this film and explain why the surface of the glass ceases to reflect the light after the film is put on. Why must the thickness of the film be equal to a quarter of the wavelength of incident light? Why must the refractive index of the film be less than that of the glass? 491. Experiments show that the luminous flux is a flux of separate photons or light quanta. Each photon has an energy E = hv where h : 6.62 >< 10*7 erg·s is Planck’s constant and v is the frequency of the light wave. Determine the energy of the photons emitted by a yellow

126 PROBLEMS sodium flame if the frequency of the yellow line of sodium is v = 5 >< 1013 s‘1. 492. When an electron was transferred in a hydrogen atom from one stable level to another a quantum of light was emitted with a frequency v = 4.57 >< 101* s”1. Determine the change in the energy of the electron in the atom due to this emission. 493. The great Russian physicist P.N. Lebedev established in his experiments that light exerts on bodies of ray absorptivity a pressure numerically =‘ “ equal to all the energy which the light brings in one second divided by the velocity of light (all magnitudes are in the N cgs system). M What force would the sunrays exert `7 on the Earth if they were entirely absorbed by the earth surface? In normal incidence, the sunrays supply 1.94 cal to each square centimetre of the Earth’s surface every minute. E 494. Lebedev’s device was used to _ measure the pressure exerted by light ‘ on the wings ofa light suspension shown Fig_ 163 in Fig. 163. Each wing has two circles one of which is darkened. Find the force with which the luminous flux acts on the darkened circle in Lebedev’s experiments if this flux supplies an energy of 1.5 cal per minute per each square centimeter of the illuminated surface. The diameter of the circle is 5 mm. Assume that all light is completely absorbed by the circle. 495. Prove that the force exerted by the light from the Sun on any body diminishes in proportion to the square of the distance of this body from the Sun. 30. Fundamentals of Photometry 496. A book can easily be read with an illumination of 50 lx. At what height should a lamp of 50 cd be hanging above a table to provide good illumination of its surface which lies directly under the lamp?

CHAPTER iv. OPTICS 127 497. The width of the aperture of a cinema projector is 1.2 cm and that of the screen is 2.4 m. How much stronger will the illumination of the aperture in the projector be than that of the screen? What should the illumination of the aperture be if the minimum permissible illumination of the , screen is 4 lx? ··-Q ? 498. A desk-lamp of height lj h = 30 cm stands on a table (Fig. 164). ·¤ Determine the illumination at a point on the table surface Q-l . a distance cz == 60 cm from the E lamp. The luminous intensity , · of the lamp is 25 cd. r$__——T_-G 499. At what distance Mg- 164 should the posts for street lamps be installed so that the illumination on the ground at the point lying halfway between two posts is not less than 4/15 lx? The height of the posts is h = 12 m. The luminous intensity of the lamps is I = 300 cd. Assume that a noti` ceable illumination is provided ;. .__ only by the two lamps on either ,. " ` Ai I n side. t i · 500. During fitting jobs in a __ 0 Ul Y subway an electric lamp is secured "° _? """""l"""" “" at the top point of the tunnel A · (Fig. 165). What is the ratio betlg J ween the illuminations produced ’· At i ii " by the lamp at the lowest point ' B and those at the point C lying Fig 165 at the level with the horizontal ' cross section of the tunnel? The luminous intensity of the lamp is the same in all directions. 501. A lamp of 400 cd is installed in a narrow-f1lm cinema projector. What illumination can be produced by this projector on a screen 3 mz in area if only 0.3 per cent of the light emitted by the lamp falls on the screen? 502. Three point sources of light are arranged at the vertices of an equilateral triangle. A small plate is placed

128 pnoennms in the centre of the triangle perpendicular to its plane and parallel to one of its sides (Fig. 166). Determine the illumination of this plate if the intensity of each source of light isI and the length ° of the side of the triangle is Z. 503. In constant conditions of illu1 mination a certain object is photographed first from a large distance and then from a small distance. ‘ ' How will the illuminations of the Fi 166 photographic plates in the camera difg' fer in these two cases? Which of the two requires a longer exposure? 504. An electric lamp of 100 cd consumes 0.5 W of electric energy per candela. Determine the efficiency 1] of this lamp if the mechanical equivalent of light is equal to 0.00161 W/lu. Calculate the quantity of light in ergs emitted by the lamp every second. 505. The sunrays which reach the earth surface bring in every minute an energy approximately equal to 1.94 cal per 1 cm2 of the terrestrial surface (with perpendicular incidence). Determine the total amount of energy received by the entire terrestrial surface. What fraction is this of the total energy of light emission from the Sun? What planet receives more energy from the Sun—the Earth or Jupiter? The distance from the Earth to the Sun is R, = 1.5 >< 10** km, the distance to Jupiter R2 is 5.20 times larger than to the Earth, the radius of the Earth is 6.3 >< 103 km and the radius of Jupiter is 11.14 times that of the Earth. 31. The Laws of Rectilinear Propagation and Reflection of Light Together with the problems on the position and the size of an image produced by various optical systems, Secs 31-34 include many problems involving calculation of the conditions when an observer can see these images. These problems require a clear understanding of the meaning of auxiliary rays in geometrical optics used to

CHAPTER IV. OPTICS construct images. It is equally important to know the methods for determining the rays that actually form these images. This will considerably facilitate the solution of such problems, for example, as the construction of an object positioned at some distance from a flat mirror, or of the image of a large object in a small lens or in a lens half covered by an opaque screen, and determining the position of an observer’s eye for simultaneous viewing several images produced by optical systems. In Secs 32-34 pay attention to the rules used to End graphically the focus of the rays in a beam that has passed through an optical system (for example, Problems 555-557). In solving these problems, study the specific uses of the equations of spherical mirrors and lenses for calculating the position of the images produced by systems with converging beams. The problems in the sections that follow should be solved in the order in which they are presented because many of them are based on the results of previous problems. 506. How should a point source, a flat object and a screen be placed for the outline of the shadow on the screen to be similar to that of the object? 507. An electric lamp is placed into a frosted glass sphere of radius 20 cm and is suspended at a height of 5 m above the floor. A ball of radius 10 cm is held under the lamp at a height of 1 m. Determine the dimensions of the shadow and half-shadow cast by the ball. At what height should the ball be placed for the shadow on the floor to disappear? What will the dimensions of the half-shadow be in this case? What should the diameter of the ball be for the dimensions of its shadow to be the same irrespective of the distance from the ball to the floor? 508. The following simple method can be used to compare the luminous intensity of two sources: a thick rod D and sources S, and S2 some distance away are placed in front of a semi—opaque screen AB (Fig. 167). The sources are so arranged that the half-shadows AO and OB are of the same luminance. 9-1218

130 _ PROBLEMS In what directions should the sources be moved so that the half—shadows cast by them are in contact all the time? What patterns will be observed when the sources are shifted in any other directions? 509. The image of an object is obtained using a box with a small aperture (Fig. 168). The depth of the box EC = 20 cm, the distance to the object CD = 20 cm and the diameter of the aperture C is d = 1 mm. B \\\\Q»*"·*"/,}.7.81 Q. //,/ "`°*•S, _ ETT;/r§'s&g.E;ZT;-E-p Fig. 167 Fig. 168 Can the parts of an object 2 mm in size be distinguished on the image in these conditions? 510. What will the shape of a light spot be if the dimensions of the mirror are small and those of the source are large? 511. In one of his notes M.V. Lomonosov poses the following question: "Any colour if moistened with water becomes deeper. Why?" The colour of the surfaces of bodies that can be impregnated with water does indeed grow darker and richer after moistening. Explain this phenomenon. 512. One of the expressions of the laws of propagation of light is Fermat’s principle assert-ing that light always propagates along the shortest paths. Consider the following case: light is emitted from a source A (Fig. 169), then reflected from a mirror and reaches a point B. Prove that the path ACB as determined by the law of reflection is the Shortest of all possible paths of the ray. 513. Two pins A and B arranged as shown in Fig. 170 are

CHAPTER rv. oprxcs 131 stuck in front of a mirror. What arrangement of the images of these pins will be seen by an observer in different viewing positions? In what position of the eye will the image of the pins be superimposed on each other? 514. An object O’O and a mirror AC are placed as shown in Fig. 171. Construct the image of this object in the mirror. Where should the eye be placed to observe the image of the entire object? 515. A desk—lamp is placed in front of a mirror. What will be the change in the distance between the lamp and its image if the mirror is drawn 5 cm away from the lamp? A B é B \ 0\ /\0" K" " s “ 0/ A\$A Fig. 169 Fig. 170 Fig. 171 516. A man stands in front of a mirror and looks at himself with one eye. What place should be covered in the mirror so as to keep the image of the other eye out of vision? 517. A ball is placed on a horizontal table. At what angle to the plane of the table should a mirror be placed to have the image of the ball moving vertically when the ball is brought towards the mirror? 518. A light ray is incident on a mirror. The mirror is turned through 1° about the axis lying in the plane of the mirror perpendicular to the ray. Through what angle on will the reflected ray be turned in this case? What distance .2: will the light spot move on a screen set perpendicularly to the reflected ray at a distance l = 5 m from the mirror? 519. A mirror 1 m high hangs on a wall. A man stands a distance of 2 m away from the mirror. What is the height of the portion of the opposite wall in the room that can be 9*

132 pnonnnms seen by the man in the mirror without changing the position of his head? The wall is 4 m from the mirror. 520. Determine graphically the positions of the eye when an observer can simultaneously see in a flat mirror of finite 0 A dimensions the image of a ° point and a section of a straight line placed with respect B to the mirror as shown in mm: Fig. 172. Fig. 172 521. When M.V. Lomonosov was attempting to increase the incendiary power of lenses he designed the device, shown in Fig. 173, and called it the catoptric-dioptric incendiary instrument. In this case A2, A2, A2,A5 are flat mirrors and B2, B2, B2, B2, B2 are convergent lenses. B4 Determine the angles at which A #M#»»· _, the mirrors should be- positioned .-~“"` V i and the minimum dimensions of 6; BJ these mirrors that will ensure the equality of the luminous fluxes E { incident on each lens. The diameB, B, ter of the lenses is d and the optiA A cal axes of the lenses B2, B2, B2, ' _ ’ B 2 form angles of ;|;45° with the Fig- 173 direction of the primary beam. 522. A point source of light and itis two images produced by two mirrors lie at the vertices o an equi ateral triangle. as az / U ·-_`.____·;__ M1-; 0- 4 Fig. 174 Determine the position of the mirrors with respect to the source and the angle between them. 523. Prove that a source and its two images in mirrors

CHAPTER IV. OPTICS 133 arranged at an angle cx to each other (Fig. 174) lie on a circle. Show the position of the centre of this circle. 524. Two mirrors are situated at an angle of on to each other (Fig. 174) and a source of light is placed in front of them. Where should the eye of an observer be placed to see both images formed by the mirrors simultaneously? 32. Spherical Mirrors 525. Prove that for spherical mirrors the product of the distances of the object and the image to the principal focus is always equal to the square of the principal focal length. 526. The distance from a glowing point to the principal focus of a concave mirror is p = 16 cm and the distance from the image to the principal focus is q = 100 cm. Find the principal focal length of the mirror. 527. Prove that the ratio of the length of the image formed by a concave mirror to the length of the object is equal to the ratio of their distances to the mirror. 528. An object is placed at a distance of 1 m from a concave mirror. Its image is one third the size of the object itself. Determine the position of the image, the radius of curvature of the mirror and its principal focal length. 529. The image produced by a concave mirror is one quarter the size of the object. If the object is moved b : = 5 cm closer to the mirror the , image will only be half the size L of the object. Find the principal focal length H of the mirror. H 530. The principal focal length { ll of a concave mirror is f and the { { distance from the object to the ; I principal focus is p. fd Q What is the ratio of the size of the image to the size of the object? S ’ 531. A small concave mirror L is Fig. 175 suspended from a thread in a mirror galvanometer to read the angles of turn (Fig. 175). A scale AA. is placed at a distance l = 1 m from the mirror and a lamp S is adjusted underneath the scale.

134 PROBLEMS What should the focal length of the mirror be to obtain on the scale the real image of the aperture in the lamp? To what distance d will the image be shifted on the scale if the mirror is turned through a small angle qa? 532. A concave mirror forms the real image of a point source lying on the optical axis at a distance of 50 cm from the mirror. The focal length of the mirror is 25 cm. The mirror is cut in two and its halves are drawn a distance of 1 cm apart in a direction perpendicular to the optical axis (Fig. 176). 0 1_ in `§EI ig? ‘**‘ iii? ... .3 -l --—-—- +. --*» i 0 if 2 Il ; g L-·—-——a———·I}·——b———; Fig. 176 Fig. 177 How will the images formed by the halves of the mirror be arranged? 533. A thin flat glass plate is placed in front of a convex mirror. At what distance b from the plate (Fig. 177) should a point source of light S be placed so that its image produced by the rays reflected from the front surface of the plate coincides with the image formed by the rays reflected from the mirror? The focal length of the mirror is F = = 20 cm and the distance from the plate to the mirror a = 5 cm. How can the coincidence of the images be established by direct observation? 534. The focal length of a concave mirror can roughly be determined by the following method: place a needle A at a distance d from the mirror (Fig. 178), then place a flat mirror P at a distance a from the concave mirror and a second needle B at a distance b from the flat mirror. Move the mirror P to match the virtual images A and B of both needles formed by the concave and flat mirrors. Knowing values of a, b and d corresponding to the coincidence of the images, determine the focal length of the

cnxprnn rv. OPTICS 135 mirror. Can these images be observed by the eye at the same time? 535. A screen S is placed a distance b = 5 cm from a circular convex mirror as shown in Fig. 179. An object KP of height h = 3 cm is arranged a distance a = 5 cm rom the screen. %·——-——·d·—-——-——··-—·—; S {AH ¤ 2P" / ll , 1B 1 l ;i}HI ¤—¤—--—-cZ—————+——1»~·-»l "·—b"""l}"""""""""‘ Fig. 178 Fig. 179 Where must an observer position himself to see the image of the entire object? What are the maximum dimensions of the object (with the given arrangement of the object, the mirror and the screen) for the mirror to reproduce an image of the entire object? The diameter of the mirror is d == 10 cm. 536. At what distance from one’s face should a pocket convex mirror 5 cm in cross section be held to see all of the face if the focal length of the mirror is 7.5 cm and the length of the face is 20 cm? 537. The internal surface of the walls of a sphere is specular. The radius of the sphere is R = 36 cm. A point source S is placed a distance R/2 from the centre of the sphere and sends light to the remote part of the sphere. Where will the image of the source be after two reflections-from the remote and the nearest walls of the sphere? How will the position of the image change if the source sends light to the nearest wall? 538. A point source of light S is placed on the major optical axis of a concave mirror at a distance of 60 cm. At what distance from the concave mirror should a flat mirror be placed for the rays to converge again at the point S having been reflected from the concave mirror and then from the flat one? Will the position of the point where

136 PROBLEMS the rays meet change if they are first reflected from the flat mirror? The radius of the concave mirror is 80 cm. 539. Convex and concave mirrors have the same radii of curvature R. The distance between the mirrors is 2R. At what point on the common optical axis of the mirrors should a point source of light A be placed for the ray-s to converge at the point A after being reflected first on the convex and then on the concave mirror? Where will the rays meet if they are first reflected from the concave mirror? 33. Refraction of Light at a Plane Boundary 540. A ray of light falls on a glass plate of refractive index n = 1.5. What is the angle of incidence of the ray if the angle between the reflected and refracted rays is 90°? 541. A pile 4 m high driven into the bottom of a lake protrudes by 1 m above the water. Determine the length of the shadow of the pile on the bottom of the lake if the sunrays make an angle of 45° with the water surface. The refractive index of water is 4/3. 542. A swimmer observes from under the water a luminous object above his head at a distance of 75 cm above the water surface. What is the visible distance of the object above the water surface? The refractive index of water n = 4/3. 543. A point source of light is arranged at a height h above the surface of water. Where will the image of this source in the flat mirror-like bottom of a vessel be if the depth of the vessel full of water is d? 544. What is the apparent distance from the surface of water to the image formed in Problem 543 by a mirror if the observer is standing in air and views the image vertically downwards? \/ 545. One face of a prism with a refractive angle of 30° is coated with silver. A ray incident on another face at an angle of 45° is refracted and reflected from the silver-coated face and retraces its path. What is the refractive index of the prism? 546. A coin lies on the bottom of a vessel filled with water to a depth of 40 cm.

CHAPTER IV. OPTICS 137 At what height should a small electric lamp be placed above the water surface so that its image produced by the rays reflected from the water surface coincides with the image of the coin formed by the refracted rays? How can the coincidence of the images of the lamp and the coin be established by direct observation? The observation is made along a vertical line. 547. The basic section of a prism is an equilateral triangle. A ray is incident on the prism perpendicular to one of its faces. VVhat will the path of this ray be for various refractive indices of the prism? 548. When bright sources of light are photographed, thick photographic plates exhibit around the images of the sources haloes whose internal boundary is sharp and whose external one is diffused. Explain the origin and nature of the haloes. Determine the refractive index of the glass plate if its thickness is d = 3.74 mm and the radius of the sharp boundary of the halo around the image of a point source is a = 4.48 mm. 549. The perpendicular faces of a right isosceles prism are coated with silver. Prove that the rays incident at an arbitrary angle on the hypotenuse face will emerge from the prism parallel to the initial direction. 550. In his notes on physics M.V. Lomonosov offers the ollowing observation for explanation: "Wet paper is more transparent than crushed glass." Explain these phenomena. 34. Lenses and Composite Optical Systems 551. When observed from the Earth the angular diameter of the solar disk is qp = 32'. Determine the diameter of the image of the Sun formed by a convergent lens with a focal length f = 0.25 m. 552. Where should an object be placed for a thin lens to produce its erect image in full size? 553. A narrow beam of light passing through an aperture in a screen S as shown in Fig. 180 is incident on a convergent lens.

138 pnonnnms Construct the path of the ray after it emerges from the lens. The position of the foci of the lens is known. 554. A converging beam of rays passes through a round aperture in a screen as shown in Fig. 181. The apex of the beam A is at a distance of 15 cm from the screen. S in ._%.i-E-·•-—·• *'* -- .... F _,,__ €,C___ *.*1 “ Fig. 180 Fig. 181 How will the distance from the focus of the rays to the screen change if a convergent lens is inserted in the aperture with a focal length of 30 cm? Plot the path of the rays after the lens is fitted. 555. A converging beam of rays is incident on a diverging · lens. Having passed through the lens the rays intersect __ __ _ - .__ ___________F at a point 15 cm from the °Z="`° " `° " """~;:__A_ lens. If the lens is removed ··""' the point where the rays I meet will move 5 cm closer B to the mounting that holds Fig- 182 the lens. Find the focal length of the lens. 556. The rays of a converging beam meet at a point A. A diverging lens is placed in their path in the plane B (Fig. 182). Plot the position of the point where the rays meet after passing through the lens. The position of the principal foci FF is known. 557. In what position of the eye and for what distance between a point source and a convergent lens can an observer simultaneously see this source lying on the optical

CHAPTER rv. OPTICS 139 axis of the lens and its image produced by the lens? The focal length of the lens is f and its diameter is d. 558. The focal length of a convergent lens is 10 cm, the distance of an object from the front focus is 5 cm and the linear dimension of the object is 2 cm. Determine the size of the image. 559. An image of a bright square is obtained on a screen with the aid of a convergent lens. The distance between the square and the lens is 30 cm. The area of the image is four times larger than that of the square. Determine the position of the image and the focal length of the lens. 560. Photographs of the ground are taken from an aircraft flying at an altitude of 2,000 m by a camera with a focal length of 50 cm. What will the scale of the photographs be? How will the scale change if the aircraft flies at an altitude of 1,000 m? 561. The size of the film in the camera in the previous problem is 18 X 18 cm. What area can be photographed by this camera at any one time? 562. A convergent lens forms on a screen an image of a lamp magnified to twice its normal size. After the lens has been moved 36 cm closer to the screen it gives an image diminished by a factor of two. Find the focal length of the lens. 563. What are the smallest details of an object that can be observed separately with the naked eye at a distance of 2 km? The minimum angle of vision of the eye is cp = 1’ 564. A thin convexo-convex lens is placed on a flat mirror. Where should a point source of light be arranged so that its image produced by this system is real and coincides with the source itself? 565. An optical system consists of a convergent lens with a focal length of 30 cm and a flat mirror placed at a distance b = 15 cm from the lens. Determine the position of the image formed by this system if an object is at a distance a, = 15 cm in front of the lens. Plot the path of the rays in this case. 566. Plot the image of an object in an optical system consisting of a convergent lens and a flat mirror arranged

140 PROBLEMS l in the focal plane of the lens. The object is in front of the lens and between the focus and the double focal length of the lens. What will the size of the image be if the object is positioned arbitrarily? 567. Determine the position of the image produced by an optical system consisting of a concave mirror with a focal length of 10 cm and a convergent lens with a focal length of 20 cm. The distance from the mirror to the lens is 30 cm and from the lens to the object 40 cm. Plot the image. 568. A convergent and a diverging lenses having focal lengths of 30 and 10 cm, respectively, are arranged at a distance of 20 cm from each other. Where should a source of light be placed for this system to emit a beam of parallel rays? 569. Plot the image of an object formed by a system of two convergent lenses. The focal length of the iirst lens is 9 cm and of the second 15 cm. The second 0 4, lens is in the focal plane of the iirst lens. 4,,* The object is at a distance of 36 cm from ah the first lens. Q Calculate the distance az of the image 4; from the second lens. 570. A plano—parallel plate is cut as Fig. 183 shown in Fig. 183, and the lenses thus obtained are slightly drawn apart. How will a beam of parallel rays change after passing through this system if the beam is incident: (a) from the side of the convex lens? (b) from the side of the concave lens? How will the behaviour of the beam depend on the distance between the lenses?

ANSWERS AND SOLUTIONS Chapter I MECHANICS 1. Rectilinear Uniform Motion 1. (1) at a distance of 8.5 m; (2) at a distance of 21.5 m. Solution. (1) The time of propagation of the sound to the man in the opera house is t, = % where S, is the distance from the stage and v is the velocity of sound. The time of propagation of the radio waves to the listener is tz = Zi! where S3 is the distance from the opera house to the receiver and c is the velocity of propagation of the S radio waves. If the listener and the man at H the opera hear the sounds at the same 250 time, then t, = tz and '%=% or ,50 i` Si = S2 •%· . { - tlzr (2) S3 = S3 -— E? where S3 == 30 0 { 2 3 m and S3 is the distance from the F1g° 184 listener to the radio receiver. 2. The cars will meet after 2.5 hours at a distance of 150 km from M (Fig. 184). 3. Eleven cars (Fig. 185). Solution. Assume that the man left B one hour after all the other cars set off. The motion of his car is depicted by the straight line BC. The lines 2, 3, 4, etc., show the motion of the cars coming from A in 50, 40, 30, etc., minutes before the man has set off in his car. The lines 8, 9, 10, etc., show the motion of the cars leaving A after 10, 20, 30, etc., minutes after the man has departed from B. Obviously, the number of the cars which the man meets en route will be equal to the number of the points of intersection with the straight ine BC. 4.S=51O m; u=85O ID./S. _ Solution. Since the velocity of light is many times greater than that of sound in the air, t, may be assumed as being equal to the time of Eight of the shell and tz to the sum of the times of flight of the shell and the propagation of sound from the place where the shell

M2 ANSWERS AND SOLUTIONS bursts to the gun. For this reason the time of propagation of sound will be tz - t, and the shell range S == v (tz -—- t,), where v is the _ , S U([2····t1) velocity of sound; the velocity of the shell is u = E- = --7;5. I = 6 S. S/1111 /00 90 80 70 B 60· ·*·—· · ······ ····*— 7 —·** " * 50 I 40 // 30 1 2/3 8 9 10 11 12 ’” J g0 A /0 20 30 40 50 00 70 80 90 /00 //0 /20 Z/nin Fig. 185 Solution. The second train will travel with a speed v = vi -|- v2 relative to the passenger. With this motion the oncoming train will travel a distance equal to its length in a time l i=-1 ”1+Uz 6. v = 36 km/bl'. Solution. If v is the speed of the electric train relative to the earth its velocity relative to the train coming in the opposite direction will be 2v and may be expressed through the length l and the time t of passage of the oncoming train, i.e., 2v = %· where l = 16.5 >< 10 —{— -|-10>< 10° a d 2.5 X 10· dyn. “‘“ H Solution. To determine the tensions, write the equations of Newton’s second law for each body separately. Both masses move with the same acceleratwns a. The forces F and f_(Fig. 203) act on rn, and only one force f on mz. The second law equations for the masses m, and mz will take the form F-j= mia, f= mza

CHAPTER I. MECHANICS 159 The solution of these equations gives us the required values a_,-:1....1;.;. and m1+m2 ’”1+m2° 3 1 il , ::3 ·-— F° 7;-*- ° ZZ-—· , Solution. Force F will cause the entire system to move with an acriielvgation a. The equations of Newton’s second law for each block W1 e F-fi= vw. fi--1‘z= ma. fz—f¤= ma. fs= ma where ji, fz, is are the tensions in the threads (Fig. 204). I is ’ n · t · fa f2 J9 777. 772 772 . nI I I Fé Fig. 204 By solving these equations it is possible to calculate all the tensions as well as the acceleration a with which the system will move. 68. Solution. It is difficult to start a heavy railway train when the couplings between the freight-cars are tensioned. In this case the traction of the locomotive has to impart an acceleration to the whole train at once. lf the train is iirst backed up, the couplings between the cars will be slackened and the locomotive can with the same {ull impart much larger accelerations first to the nearest car and t en, successively, to all the other cars. 69. Solution. lf before the motion begins all the couplings in the train are tensioned, the break may occur in the couplings of the cars closest to the locomotive. The tension in these couplings should be greatest since it is intended to produce an acceleration for the greater mass of the oars behind all at once (see Problem 68L. If all the couplings between the cars are first slac ened the break may occur in any Fart of the train depending on the ratios of the tensions in the coup ings between separate cars. 70. The dynamometer will show a force: (1) in ~f= 1 1121: (2) fn iv F= 2 kefi ¤¤d (3) ;n—.:£?]:=1.5 kgf. Solution. In all three cases the system will move with some acceleration a in the direction of the larger force and the dynamometer will show the bonding force fn acting between the weights. To hud fn, it is necessary to write the equation of IjIewton’s second law separately for each weight. For the first case (Fig. 205). F··fn= Ma. fn-—f= ma

Hence, a—---L -M ·-|- m and M fn ·— F-·· HQFE (F ··}) Since m < M and 1 it may be assumed that M+ m fn W f The other cases can he considered in a similar manner utilizing J, the equations of Newton’s second _, mL,. F law and the given ratios of masses. i il ’ 71. a=!tB?. g. D ' Q P +0 ° Tn, • • • • • • M P =·——- 1 k . F' 205 I P4-Q ( + ) 1. g Solution. The bodies P and Q move with accelerations of equa magnitude a. The body P is acted uson by the force of gravit and the tension in the thread f, and the bo y Q by the tension in the thread and the force of friction j, = kQ. The equation of N ewton’s second law for the motion of each body will be PQ P- =—— , —-k =-—Ig4fQga The solution of these equations determines the values of a and j __P-kQ __ PQ 1 4- ,,+0 z and 1‘—————·,,_,_Q( +k> .. mg . 72. a-. 2M +m , _ 2M (M -l-m) . T- 2M -|—m. g' 2Mm { :.50-[$7; g, F - 2T Solution. All the bodies in the system move with accelerations of equal magnitude a (Fig. 206). T e left-hand weight M is acted upon by the force of gravity Mg and the tension in the thread T, and the 1·1ght·hand weight by the force

CHAPTER 1. MECHANICS 15{ of gravity Mg, the pressure of the small weight f and the tension_in the thread T. The small weight m is acted upon by the force of gravity mg and the pressure f exerted by the /, / J, weight M. For each of the three masses the equations of N ewton’s second law will be T — Mg = Ma Mg + f — T = M a mg — f = ma The solutions of these equations deter- T la mine the values of a, T and f. _ . The pressure on the axis of the pulley ` will be equal to twice the tension in the Mg M threads F = 2T. T mi]: 1;-.. .:1/f‘&.fri’2?a.»o.oS. _ l (P 2-P 1] 8 _ mg Note. The acceleration of the weights ,- M can be found from the equations of New- My ton’s second law (see the solution to Prob- _ . P2-P, Fig. 206 lem 72) and will be oz -17:*3- g, and the 4z. time of motion from the kinematic equations of uniformly accelerated motions from rest will be 1/ W t= — a 74. The centre of gravity moves down with an acceleration j:(Pr-—Pz)2 g (P1+P2)z Solution. After a time t each weight will be dgsplaged from the t ..... initial position by a distance S = Lg- Wh61‘6 a = ·pi·j|_·Ii· 8 (SBB the solution to Problem 72). _ , In this case the centre of gravity of the system should obviously move down a certain distance L from the_1n1t1al position towards the larger weight (Fig. 207). Upon determining the centre of gravity, the distances of the centre of gravity from the weights should be inversely proportional to the magnitudes of these weights, 1.e., S -4- L ___ lh S -· L __ P2 11-1218

162 ANSWERS AND SOLUTIONS or E-$-1 P2 P,—P2 1 Pl+P2 P2 . atz Inserting the value S=-2- , we get _·Pi—P2 (liz L*Pi+PzX 2 Comparing this result with the formula for the path of uniformly accelerated motion and then inserting the values of the acceleration of the weights we find that the centre of gra* - · f vity Should move down with an acceleration i=%E ¤= (i$%)2g # The magnitude of the acceleration of the I centre of gravity is less than that of each · weight separately. _ . 75. In the first case the system moves w1th | an acceleration a == 9 cm/s“ and the force of Q M fricggn petween the block and the cart is f= v,), the boy should develop a greateipfower in the second case. 105. v, = 1 s; A, = 15 kgf-m; A2 = 37.5 kgf-m; N, = = 10 kgf—m/S and N2 = 25 kgf-m/S. Solution. In both cases the man imparts the same acceleration a = Tg- to the boat A and therefore the velocity of the boat A (v, = 1 -.: at = % t = 1 m/S) will be the Same in both cases. In the first 12 case the work done by the man is A, = % = 15 kgf—m and in the second case 22 A2 :2%+2%;:37.5 kgf-m where v2 = -’§·t is the velocity of the second boat by the end of the z third second. The power developed by the man at the end of the third second in the first case is N, = Fv, = 10 kgf-m/S and in the second N2 '—= F (U1 + U2) = kgf*H]./S if N2 > N,. 106. Solution. The velocity of the body will be less in the second case since the body’s potential energy at a height h is expended in the first case to impart kinetic energy to the body alone, whilst in the second case it IS used to impart kinetic energy to the body and the prism at the same time. 107. tan Br-= tan oz. Solution. Let us denote the velocity of the prism (Fig. 214) by u, the horizontal and vertical components of the load velocity v relative to the earth by vx and vy and the angle between the direction of motion of thB load and the h0l'iZOIlt.&l B, assuming t Dy 1 an B- vx < >

CHAPTER I. MECHANICS 173 `. Since the prism is acted upon in a vertical direction by the reaction of the support in addition to the load, the law of conservation of momentum may be applied only to the horizontal components of the velocity of the load and the prism when the behaviour of the "loadp)rism" system is considered. The velocities u and vx will obviously e linked by the ratio • Mu =mvx (2) Let the load be situated at the point A of the prism at a certain moment of time (Fig. 215). During the first second a ter this the prism v RZ `~—` · = n \\ ua A ve ”é I Y (y vg iv l \ UV “ ¥ IL x J /1'///.;./·’>/i," 7///71 /1 . /,11, /1/ ///2//1 /// Fig. 214 Fig. 215 has moved u cm to the left and the load has been displaced vx cm horizontally to the right and by vy cm vertically. All these displacements should be such as to return the load again to the prism at a certain point B. Therefore, the velocities u, vx and vy should satisfy the laws of conservation of energy and momentum and also the ratio Uy =-‘ t3I1 CZ This ratio expresses the condition that the moving load is always located on the prism. We fund from (2) that u : § vx. Inserting the value of u into equation (3), utilizing (1) and performing simple transformations we get Uy m-{-M Z **' ;‘*'1""" t tan B vx M an or As we would expect, tan B > tan on and B_> oc. _ _ The velocity of the load down the moving prism is directed at a larger angle to the horizontal than during descent from a stationary prism. Using the law of conservation of energy and knowing the height of the initial position of the load it is possible to calculate the velocities u and v. _ _ 108. After the impact the balls will exchange their velocities. Solution. If the masses of the balls are denoted by mi and mz and the velocities after impact by as and y, we can obtain from the law of conservation of momentum maui + mzvz = mw? + mzy (1)

{74 ANSWERS AND SOLUTIONS Applying the law of conservation of energy and assuming that the total kinetic energy of the balls does not change after the impact we may write m,v§ m2v§ __ m1.1:2 mzyz 2" ‘+"_‘2 ‘ _f +" 2 ‘ (2) Solving equations (1) and (2) simultaneously and using mi = = mz = m, we get y = v, and a: = v2, i.e., perfectly elastic balls of equal mass will exchange their velocities after impact. If the first ball moves from the left to the right with velocity v, before impact, it will move in the opposite direction with velocity vz after impact. 109. M2 = 300 kg. Solution. Momentum before meeting ’ after meeting First boat (M 1-|-m) v M iv Second boat -M2v -(M2-|-m)v2 The momentum of the two boats before the load is shifted is equal to their momentum after _the transfer of the load, i.e., (M1+ m)v···MzU= M1v“(M2+m)v2 Hence, M2= =300 kg. U—U2 The energy of the boats before they meet is E (M1+Mz+m)v2 1:-__—'-2_-—— and after reshifting the load the energy of the boats is __ M10? (Mz+m) vi E2 ··~ T + -5* _ The energy has diminished due to the transfer of a part of the energy 1Dto heat when the velocities of the load and the second boat become the same. h 110. N=—”%-==49>< 1011 erg/s:/490 kW. 111. F = 4,896 kgf. Solution. The useful power N2 = kN, = Fu. Hence, FNI Ng=—i;Z— and F:-§;%4,896 kgf

CHAPTER I. MECHANICS 175 112. The power N == 4.2 hp. Solution. The force of friction F between the shoes and the shaft is determined from the equilibrium condition of the lever in the absorp— tion dynamometer: the moment of the force F is equal to the moment of the force Q: Fr = Ql and hence F: r The velocity of the points on the surface of the shaft will be v = 2:lWr where v = £ is the number of shaft revolutions per second N=Fv=·-g-Q-:?;£=2:n;Qvl at 4.2 hp 113. N = 11.8 kgf-m/s. H4. The maximum power should be developed by the engine at the end of the take-off run and equal to N = Fv where F is the tractive force of the propellers which, as is given in the problem, remains constant during the entire time of the run. The force F is determined from the equation of Newton’s second law for the take-off run: F —- kP = 5- (1 8 . v2 The acceleration azg- and therefore Pvz and the power is Pv3 H5. N z 1 hp. Solution. The force of friction overcome during machining is F, = kF, the velocity of the rim of the stone is v = rt dn and the required power N = Fiv = lcF at dn H6. fz 490 kgf; F z 980 kgf. Note. The pulley will rotate under the action of a force equal to F —— j. The velocity of the pulley rim IS v = 2m·n and the power N = (F —- j) 2nrn. Since it is given in the problem that F = 2f, then NN = ,.;,_, F—·.:—-— f 2:rtrn and Jtrn

176 ANSWERS AND SOLUTIONS 8. Dynamics of a Point Moving in a Circle 117. A0 = AH; = 25 cm; OB = 75 cm. Solution. The centripetal force acting on M will be F1 = co2Mx (Fig. 216). The force acting on m will be F2 = mzm. (l — x). From the given conditions F, must *___I l_x be equal to F2, i.e., ` I co2M:1: = mzm (I —— sc) M ” # m °" A- T __ __ ml A » ,2 8 x..AO---—-M+m Fig. 216 These expressions also determine the distances from the balls to the centre of gravity of the system. The tensions in the threads are the same when the centre of rotation coincides with the centre of gravity of the system. _ 118. co= I/%=7 s·1. Solution. The balls will be in equilibrium when the centripetal force acting on the ball B is equal to the weight of the ball A, i.e., when rmzm = mg. Hence, on = The equilibrium will be unstable, __ ue. vo > 1/gn. Solution. The velocity of the washer vo should be such that the parabolic path it takes under the force of gravity can pass outside the hemisphere, only touching it at the I upper point A (Fig. 217). Ai U When the washer moves along a pa- _ ° rabola its vertical acceleration at the `\ point A will be g and the centripetal \\ acceleration for motion in a circle of x 2. radius R withavelocity vo will be ' ' 2 F' . 217 If g Q l% the curvature of the pa- lg rabola will be less than the curvature of the surface of the hemisphere and the parabola will be outside the hemisphere, i.e., the washer williiot slide over the hemisphere at velocities vo when vo ,> > Vail- 4 ZR at 120. a:-f¥—=0.033 m/sz (T is the time for one complete revolution of the _Earth). The reduction 1n the weight of bodies on the equator caused by the rtitation of the Earth comes approximately to 0.0034 of the force of a raction.

CHAPTER I. MECHANICS 177 \/ 2gh _ cozllz 121. TZ—· Solution. (1) If the particles of water emerge from the pump with velocity v they can rise to such a height h at which all their kinetic energy will pass into potential energy, i.e., the following will alwayg be true: v2 = 2gh Assuming that the velocity of the water particles is equal to the linear velocity of the ends of the pump vanes it is possible to determine the number of revolutions n o the pump n _ v = VE `_ 2:tR 2:rtR (2) When equilibrium is established and the water rises in the pipe to the maximum height, the pressure at the exit of the pump will become equal to the weight of the water column of height h, i.e., 22S P: gh = % = 22* R2 I R where oi = 2rcn is the angular velocity of the water particles in the pump. Inside the pump, during motion from the axis to p D the ends of the vanes, the pressure will grow in pro- Z " portion to the square of the distance to the axis. (3) To calculate the centripetal force, let us separate a thin layer of water between the cylinders of radii R and r (Fig. 218). The thickness of the layer Fig. 218 S = R - r should be small enough for the velocities of all the particles of this layer to be regarded as identical. Each element of the water volume supported on 1 cm2 of the surface of the internal cylinder r will be acted upon by a force equal to the pressure difference 2 2 2 2 2 F;-p,..p2=.°l;{..-.£°é°.:_£.°i. (R2-,-2) As the mass of the water in this volume is m = dS >< 1 = R —_r (where d = 1 is the density of the water) and assuming (since S is small) that R + r = 2R, we get 22 F=£-of (R-}—r) (R-r)=—T;— (R-|—r) m meo2R i.e., the (pressures are distributed in a centrifugal pump so that the pressure ifierence acting on each layer is enough to produce the required centripetal accelerations of the water particles present in this layer_ oaznpnz E 4 122. Fir:-—-;-:0.08 kgf; co:-l/-7-2-:2.2 -7;12-1218

178 ANSWERS AND SOLUTIONS Solution. The centripetal acceleration for a moving load is only provided by the force of friction and in the first case 22 F,, Z mm2R :4:rt2Rmn2 Z £gl The load will begin to slide with that angular velocity at which the centripetal force becomes equal to the maximum force of friction at rest, i.e., when kP = m0)2R or _ kP __]/H °’ · l/H · w __ lofo . _ mwzfolo 123. RZ f0__mm2 , FZ f0_mwg . Solution. Assume that the length of the cord has increased by l, cm Then the radius of the circle along which the ball will move is R = = lo + l, and the tension in the cord is F = fol,. When the ball rotates with an angular velocity co it will have a centripetal acceleration m’R = of (lo -{- l,) and by Newton’s second law F = mo>2R. Inserting the values of F and oJ2R in N ewton’s second law equation we obtain foli = mwz (lo + Z1) or I — ?Tl0)2l0 1 — f0—m2 Hence, _ _ lofo R-lo+l1 — f0_mm2 and _ __ mwzfolo Pvz Pvz • F 1 ° z —·;—· ‘ ; ——· , 124 1 P, F2 P gR , F3 P-}-gR Solution. The car is acted upon in the vertical direction by two forces: the weight P and the reaction of the support F. (1) When _the car runs over a horizontal Hat bridge there are no acceleratmns 1n the vertical direction and the sum of the forces acting on the car 1n th1s direction should by Newton’s second law be equal to zero P—F1=O01‘Fi=P _ (2) When the car runs over a convex bridge a centripetal accelera1:1011 will act vert1cally downwards and therefore mvz P·F2:T

CHAPTER I. MECHANICS 179 or p .2 Fz=P—Ei/R-. Fz \ B 2`\/4l2—-L2 2 2 Solution. The weight of the rod P P acting through the centre of gravity of the rod may be replaced by two equal Fig. 231 forces gapplied at the ends of the rod. In the first case (Fig. 231) the force % applied to the rod at the pointA should be resolved into two components: F,-—in the direction of the extension of the rope AC, and F2—acting along the rod and directed towards the centre of gravity of the rod. As the triangle of forces and the triangle AOC are similar we get for the tension in the rope T, and the force F2: Pl 2]/,2 L2 1/412-1.2 -7

CHAPTER 1. MECHANICS 189 I/ L2 2 1/412-L2 ;2__. 4 The force F2 and the force Fg applied at the point B compress the rod with a force of 6 kgf. In the second case the tension in the rope will be the same as in the fiillstfcase. The forces Fg and F2 will extend the rod AB with a force o6g._ 149. At or > 120°. M l B 150. Q = 200 kgf. , [ Note. The magnitude of the force is determined H from the similarity of triangles OCB and OKM [ i /“ I (Fig. 232) and is equal to Q = Q; . [ { /4 Q- C i 151. T = —§ cos on. When oc changes from 0 to `xx E x\ 90° the tension in the rope T diminishes from `\\% \\ ig- to zero. N \ A Note. The ma`gnitude of the force T is determined from the condition of equilibrium of the board. --:2.,; The sum of the moments of the forces T and P with respect to the point A should be equal to zero, i.e., Tl=P L cos on where l = AB is the an 2 P Fig. 232 length of the board; hence, T.: Tees oz. 152. The weight of the beam is Qi: 300 kgf. Solution. The weight of the over anging end of the beam equal to % acts through the point 0 (Fig. 233). The equation of the moments of the forces with respect to the point C ,___3l____,___|l will be I 3 { if 3 3 z Q 1 ' 1”'” ‘2;Q> M3 -|- M3 and therefore the left-hand Eau will move down. 155. T e tensioning force is Px = P cos oz. z 86.6 kgf. The deflecting force is Py = P sin on = 50 kgf. 156. The length of the overhanging edges of the bricks will beg- , é- , écounting from the top brick. _ Solutions. Since the bricks are homogeneous the point of application of the weight of each brick will lie half way along its ength. Consequently, the first top brick will still be in equilibrium with respect to the second when its centre . ci; -3 of gravity is arranged above the __? ‘ ' ‘ edges of the second brick, i.e., the ma; ' p ximum length of thc; free edge of the j_.;;g§.j.·¤_.; first brick will be -2- . p / P The centre of gravity of the first I 2 and second bricks taken together will be 3P at a distance éfrom the edge of the Fig. 234 second brick. This is precisely the length by which the second brick may be displaced with respect to a third. The centre of gravity of three bricks lies on the line AC and its position can be determined from the equation P (-g-:) = 2P:c (Fig; 234) from which we get x = , i.e., the third brick may jut out over a fourth by not more than % of its length. 157. The equilibrium will be disturbed, but can be restored by applying to the right—hand end of the beam a force F = ig- equal to the weight of the part cut off. 158. Solution. The magnitude of the force F can be found from

CHAPTER I. MECHANICS 191 the equation of the moments of the forces relative to the bottom of the ladder Ph tanot=-Ilcosoc, F:-EEX-gl-ql2 l coszot 159. The force required in the second case is half that required in the first. 160. F = 1.4 kgf. Solution. For the block to be in eqiuilibrium on the inclined surface the force of friction f,. = kN shoul be equal to the component of the weight directed allong the inclined surface (Fig. 235) h P2=·'P fr S , _ 1/ l2—h2 . The normal pressure N =F-{-P —-—€- . Inserting the values of N, P2 and ff,. into the condition for equilibrium we get z_. h k(P—-—V’l ’“+F)=:-P il ~ pl F=%(%P-%?v%:¤) 161. F=-gl-:50 kgf. /1 Solution. In order to raise the log to the P height h each rope should be pulled up a dis- F. 235 tance 2l. On the basis of the "golden rule" lg' of mechanics Ph. = 2 (F X 2l), where Ph is the work done by the force of gravity and F >< 2l is the work done by the force of tension in one rope. Therefore, F == Q? . The problem can also be solved considering the equilibrium of forces applied to the log. 162. F = 2.5 kgf. Note. The force required to keep the differential winch in equilibrium can be determined from the principle of the moments or from the "golden rule" of mechanics. From the equation of the moments fg rz —|— Fl =-· ri -1% from which __ P (Ti —·· T2) F` 2l 163. III. P

192 ANSWERS AND SOLUTIONS Note. The force F can be determined by the "golden 1·ule" of mechanics and from the condition of equilibrium of forces. If the end B is attached a distance l from the centre of gravity of the log, the end C should be fastened a distance 2l from it. In this case the point of application of the resultant of the tensions F and 2F in the ro es will coincide with the centre of gravity and the rising log will be in a horizontal position. 165. mz = (mz — mz) sin ci; N = (mi — mz) g cos oi. Solution. The body of mass mz is acted upon by its weight P, = .= mig, the tensions in the threads mzg and mzg and the reaction N of the inclined surface. The equations of equilibrium will have the form: (m, — mz) g cos oi == N and (mz — mz) g sin oi = mzg from which mz = (mz — mz) sin a, N = (mz — mz) g cos ci 166. Q = 15 kgf; oi z 56°. Solution. If the system is in equilibrium, the resultant of the forces P and M applied at the point A should be equal in magnitude to the force Q, i.e., Q= \/ P2-—M2 z 15 kgf cosoi==%, cz»·¤,·56° The problem can also be solved in a different manner. Considering that the sum of the projections of forces in any direction should be equal to zero we may write / I Pcosoi=M, Psinoi=Q I ° from which Q and oi can be deter\ U6 mined. \g·{ 167. Q=§— lf the point A is shifted, the equilibrium will be disturbed. The weight P will go down B» P and Q will go up. Q ( Note. This result can easily be A·| I obtained from the "golden rule" of 0 A mechanics or when the sum of forces acting on the movable pulley is con0 sidered. 168. Q = 3P = 9 kgf. Fig. 236 Solution. For the system to be in equilibrium the moments of both forces P and Q should be the same, i.e., M, = Pl = Mz = Q é and therefore Q : 3P. If the rod is deflected upwards from the position of equilibrium through a small angle oi (Fig. 236), the moments of the forces Q and P

CHAPTER I. MECHANICS 193 will no.longer be the same. After the rod is turned, the moment of the force Q will be M2 = Q é- cos oz. Obviously, M2 < M2. It is easy to see that this change in the moment M2 is due only to the change in the direction of the rod. When the rod is turned through the angle on the direction of the force P will, also be changed by an angle B and so the moment of the force P after the rod turns will be M; = Pl cos (ot —|— B). The value of M { is affected by both deflections (of the rod and the thread) in a similar way and the reduction in M, with a given deflection of the rod will therefore always be larger than the decrease in M2. 'l`he resultant moment M2 -— M, ·;& O will cause a clockwise rotation of the rod. The rod will tend to return to the horizontal position corresponding to a stable position of equilibrium. By considering the change in the moments M, and M2 when the rod is displaced downwards and the change in the direction of the force P it can be shown that if the rod is displaced downwards, it will tend to return to the position of equilibrium. 169. The centre of gravity lies in the middle of the bisector of the angle on whose vertex is situated the ball of mass 2m. 170. The centre of gravity will be at a distance x___ Rrz _ 2 (R2-—r2) from the point O. Solution. The weight of the disk before the hole is cut out may be represented as the resultant of two forces: the weight of the cut-out portion and that of the remaining portion, each of which is applied through the centre of gravity of the res- 8 0 A sective part. This makes it possible to re- mWAW, W&__,,,, uce the problem of finding the centre of ` gy E gravity o the intricate figure left after the 2 P hole is cut out to resolving parallel forces ’ and finding one of the components of the [iv forces from the given resultant and the oth- P er component force. The weight P of a uniform solid disk is Fig, 237 proportional to R2 and acts through the centre of the disk O. The weight P, of the cut-out portion of the disk is proportional to rz and acts through the centre of the hole A (Fig. 237). The weight of the remaining portion P2 equal to the difference P — P, acts through a certain point B at a distance x from O. It follows from the rules for summation of parallel forces that the distances as and ig- of the points of application of the forces P, and P2 from the point O should satisfy the ratio at P, R P2 2 13-1218

194 ANSWERS AND SOLUTIONS Knowing that 2;:.}];.:...i. P2 P-··P{ .R2—• T2 we obtain 2::: __ rz R — R2-rz or __ Rrz I- 2 (R2—r2) 10. Universal Gravitational Forces 4 171. yzasc X 10*10-1;;-%;;. 172. The difference in the lengths of the threads should be equal to l as 3 m. Solution. If the difference in the lengths of the threads is assumed to be equal to l and one of two identical loads is right on the terrestrial surface, the following exlpressions can be obtained for the forces of attraction acting on the oads by the Earth: Mm Mm P1=‘Y Tg- » P 2 Z Y m where M = E; pR’ is the mass of the Earth, R is the radius of the Earth; p is the density of the Earth and m is the mass of the load. The difference P2 —— P, will be equal to the weighing error and will bg p,-p2 = E Ypmgs _ 3 R2 (R—|—l)·‘* Since l < R, the term lz which is small compared to 2Rl may be neglected in the numerator of the formula obtained and R + l z R may be assumed in the denominator. We get P,-pz Z % ypmzz and therefore I ___ 3(P1···P2) _ 821Ypm 173. F sz: 4.1 >< 102** kgf. Solution. Since it is necessary to determine the mean force of attraction, assume that the Earth rotates around the Sun in a circle of radius R. In this motion the centripetal acceleration of the Earth provided by the universal gravitation force will be “:i`“` 122*2

CHAPTER I. MECHANICS 195 By Newton’s second law, 4112MR F=Ma =T where M =%:m·3p is the mass of the Earth. Inserting the value of M we shall obtain the following expression for the force of attraction of the Sun: 16:rt3 r3Rp F·TXT 174. oa z 1.3 X 1O‘3 s‘1. Solution. The weight of the bodies on the surface of the Earth will become zero at that angular velocity of Earth‘s rotation co for which the centripetal acceleration cozr corresponding to this angular velocity is equal to the acceleration of free fall of the bodies g, i.e., when cozr = g, where r is the radius of the Earth. Hence, co:]/E xs 1.3 X 10*3 1rs The value of the required angular velocity can also be obtained directly from the law of universal gravitation and Newton’s second law. The gravitational force exerted on a body by the Earth is F = =-?yprm where p is the density of the Earth and m is the mass of the body. When the weight of a body on the surface of the Earth becomes zero the equation of Newton’s second law for a body rotating with the Earth with an angular velocity co will have the form 4:11 2 T yprm = mm r and therefore _. ,2 N -2. 1 oJ..2]/ 3 vp~1.3> N2 and therefore fi > f2. The resultant of the forces ji and fz will be i= kP if- directed towards the position of equilibrium. The board will tend to go back to the position of equilibrium. Thus, the forces of friction will cause the board to oscillate. 12. Hydro- and Aerostalics 186. h 4:,410.34 m. Note. The water will rise with the piston until the pressure produced by the weight of the water column becomes equal to the atmospheric pressure. 187. P0 = 21.5 kgf/cmg. 188. F = 1.5 kgf. 189. P = 1.04 kgf/cm2. 190. d = %— do == 0.83 g/cm3 (do is the density of water). z 191. h = r. Solution. The pressure exerted on the separate elements of the side of the vessel (as measured by the height of the liquid column) changes from 0 to h in proportion to the distance of these elements from the free surface of the liquid. For this reason the total force of pressure on the side surface can be calculated from the mean pressure equal to Q- The force of pressure on the side surface will be proportional to 2nrh g- , the force of pressure on the bottom will be proportional to nrzh. The required result can be obtained by equating these forces. 192. lt will not. _ Solution. The pressure in the tube at the level of the tap A will be below atmospheric pressure. Therefore, the atmospheric pressure will not allow the water to flow out when the tap is opened. A1r will enter the tube through the tap until the atmosplheric pressure is reached inside the tube and until the water sinks to t e initial level. 193. 148.5 cm Hg. 194. h m 85 cm.

202 ANSWERS AND SOLUTIONS Solution. The water in the tube will rise until the pressure of the water column that is being formed equalizes the pressure produced by the iston. The pressure exerted by the piston is ~ gf/CII} I) The height of water column can be found from the equality dgh = P 195. P = 4 cm Hg. Solution. The heights of the layers of water and mercury hi and hg can be found from the ratios hi + hz = ho and h1'Y1 = h2Y2 where yi and Y2 are the specific gravities of water and mercury. The pressure in centimetres of mercury column can be found from the ratio P = I-’§—’L+ hz Z uz 196. h w 3.7 cm. 2 Solution. When the water is poured in, the mercury level will sink a distance h in the first vessel and rise by the same amount in the second vessel. The lpressure of the mercury column of height 2h thus formed will be equa ized by the pressure built up by the column of water and the blody floating in it, i.e., in the case of equilibrium, the following will old __ P+p where d is the density of mercury and P is the weight of water. _ P+p 197. hi = hg % : 18 cm. 1 198. hg = 0.3 cm; h, = 4.8 cm. Solution. If the displacement of the mercury levels in the rightand left—hand vessels is denoted by h, and hg (h, —|- hg = x) and the pressure IS measured in centimetres, the condition of equilibrium of the liquid will take the form hd n, 4. hz 3- .952 where do is _the density of wate1· and d is the density of mercury. As liquid is incompressible Sihi = Szhz

CHAPTER I. MECHANICS 203 where S, and Sz are the cross-sectional areas of the vessels related, from the given conditions, by the ratio Sz = 16S,. The first equality determines the condition of equilibrium of the liquids in the tube and the second expresses the constancy of the volumes of mercury transferred from the eft—hand limb to the right-hand one. From these equations: _ hodo _ 16h0d0 WW and h*" mz 199. hz z 0.6 cm. Solution. When water is poured in, the level of mercury in the narrow limb will sink to a height h, and in the broad one it will rise to a height hz = E3! The height of the water column will be l -|- h, and the height of the mercury column equalizing the weight of the water column will be h, —|— hz. Equilibrium will be established when the following ratio is observed do (Z + hi) = d (hi ·l· hz) where d is the density of mercury and do is the density of water. Hence, ldo h2"2H- 3,1,, 200. The difference in the heights of mercury levels is h, = 0.5 cm. Solution. Since we are given that both limbs have the same height the equal columns of kerosene above the water level may be ignored. The mercury level in the limb containing water will obviously be below the mercury level in the other limb (since the specific gravity of kerosene Ez is smaller than that of water yo). If the di erence in the mercury levels in the two limbs is denoted by h, the condition of equilibrium of the liquids in the tube may be written as h0'Yo = h1'Y1 + (ho — hi) Y2 Hence, YO'—Y2 h = —-— h 1 Y1‘"'Y2 0 201. 50 gf. 202. V rs, 75 dm3. 203. d .¤ 1.5 g/cm3. g()1,_ V1:}/.P.E’.; Vzzy ILIE. Y2—‘Y1 Yz‘fY1 _ Solution. Let us denote the fract1on of the volume of the ball in the upper liquid by V, and the fraction in the lower one by Vz. Then, V = V, —{— Vz. _ Each of these parts of the ball is acted ugon by the force of gravity V,·y and Vzy and the buoyance (Archime es force) V,·y, and V2?. Since the ball is in equilibrium on the boundary of the liquids, t e

204 ANSWERS AND SOLUTIONS sum of all these forces is equal to zero, i.e., (V1 + V2) Y = V1Y1 + V2)’z Hence, V'? = VIYI + (V —— V1) Y2 or v,:V.Y2;.Y. 'Y2·'Y1 Similarly, V2:-.V.JL.X!.. 'Y2‘_Y1 These formulas can be verified by the method of passage to the limit. (1) Suppose that the specific gravity of the ball is equal to that of the; uplper liquid, i.e., Y = Y,. Introducing Y = Y, into the expression or 2 we ge 'Y2·Y1 i.e., the ball is in the upper liquid. The same result will be obtained if Y = Y, is inserted in the expression for V2 VZZZV Y1_Y1 :0 Y2”Y1 (2) Suppose that the specific gravity of the ball is equal to that of the lower liquid, i.e., Y = Y2. We get: V, = 0 and V2 = V, i.e., the ball floats in the lower liquid 205. Y;-.Y.%3@..=7.25 gf/cm3. Note. Since V,=V2 (see the solution to Problem 204), then V Y2"°Y ZV Y`_°Yi Y2_Y1 Y2"'Yi or v2——v=v—vi. from which 2v=v2+v1 or v=2?-ig-Y-L 206. 0.19 of the volume. Solution. It follows from the given condition that the weight of the body IS = 0.25 VY where V is the volume of the body and Y the specific gravity of mercury. If x is the volume of the body left in the mercury after the water is poured in, the condition of equilibrium of the body may be written in the form :cY + (V —-— x) Y0 = 0.25 VY where Y0 IS the specific gravity of water. Therefore, 2; : V-_:9_4gv Y···Y0 207. d w 2.5 g/cm’.

CHAPTER I. MECHANICS 205 208. d = 1.5 g/cm3. 209. The pan with the piece of silver on it will move down. 210. V = 13 cms. 211. V z 59 cms. 212. P z 10.9 gf. Note. If P is the weight of the mercury the following equality should hold P P,—-P __ P,-——P2 Y1 + V2 _ Y0 where yo is the specific gravity of water equal to unity. 213. d = 1.8 g/cms. 214. AU, = Vgh (d — do); AU2 = 0. Solution. The body moving in water is simultaneously subjected to the force of gravity and hydrostatic forces. The work done by the hydrostatic forces, as well as the work done by the forces of gravity, does not depend on the path. We may therefore introduce the concept of the potential energy of a body acted upon by hydrostatic forces. When the body is raised to a height h its potential energy will be increased by Vdgh by the action of the forces of gravity and decreased by -—— Vdggh by the action of the hydrostatic forces. The total change in the potential energy of the body will,be If d > do, then AU, > 0 and the energy of the body increases. If d < do, then AU, < 0 and the energy of the body diminishes. When the body moves up to the height h a volume of water V is displaced downwards by the same distance. In this case the potential energy of this volume in the field of the forces of gravity will diminish by Vdogh and the energy due to the hydrostatic forces will increase by Vdogh. Therefore, the total potential energy of the water will remain constant: AU2 Z 0 P1 215. P=P,-\—v0 (V-——) :440.6 gf. Y1. . . . 216. When the set of weights IS made of material having the same density as the body being weighed. 217. v, = 1.94 gf/l; V = 1]; P0 = 125 gf. _ Solution. The following ratios can be obtained for the specific gravities of air vo, carbon dioxide Y, and water Y2 P —P P -P P ——P Yo=‘L,%)·» vl==·J—,r£. v2=-—"¥T,—£Hence, the formulas for the weight and volume of the vessel and for the specific gravity of carbon dioxide are P0:P1Yz—·I’s\’o, V: P3--P, 1 m=(pz-—P,) vz+(Ps~·P2Wo v2"'Y0 Y2—'\’o P3""Pi

206 ANSWERS AND SOLUTIONS 218. 79 kgf; 0.5 g/l. 219. The ratio between the volumes of water and alcohol should be 8 : 13. Note. The density of the mixture can be found from the relationship V1d1+Vzdz d i•*'"" -·•· 1 ° K ‘ ’ where Vi and V2 are the volumes of water and alcohol; d, and dz are their respective densities and K = 0.97 is the coefficient of reduction in the volume of the mixture. The numerator and denominator in relationship (1) determine the mass and the volume of the mixture, respectively. Hence the ratio of the volumes of water and alcohol is _ll_ Kd0—d2 V2 __- d1·—Kd0 d 220. 77.4 parts by volume of air are needed for 100 parts of carbon ioxide. Solution. The specific gravity of the mixture should be such that the weight of five litres of it is equal to the weight of the ball and the air. The weight of the ball and the air is WV -{- P. If W is the volume of the air in the mixture, the weight of the mixture will be yiW + Y2 (V — W). h The condition of equilibrium of the ball may be written as _-{nm but “- Y1V+P=Y1W+Y2(V—W) gha j E Hence, P · - ' wzv ——- S 2.181 + Yi-FY2 and 8 Fig- 243 .2;.-2;. .. . V___W -- 2.82 N 77.4. 100 221. The water level will perform periodic oscillatory motions (Fig. 243). Solution. At first the level of water will gradually rise to the hg. After reaching the height hg some of the water will be drained through the _s1phon. As soon as the entire cross section of the to of the siphon pipe is filled with water, the water level will begin to drop since from the given conditions the velocity of the water flowing from pipe IS h1gher than from pipe A. The evel will continue to sink until it reaches the height hi coinciding with the edge of pipe B. After this the process will be repeated.

CHAPTER II. HEAT AND MOLECULAR PHYSICS 207 Chapter II HEAT AND MOLECULAR PHYSICS 13. Thermal Expansion of Bodies 222. The clock will lose 1: = 8 s. Solution. A definite number N of oscillations of the pendulum will correspond to one full revolution of the hour-band. If the clock is accurate, these N °oscillations are performed in twenty-four hours. From the given conditions N Z 24 X 60 @0 2:1: I/Bl. 8 When the temperature changes by t degrees the length of the pendulum will be l = lo (1 —{— at) and the period of oscillations of the pendulum will change by T-—-T0=2J1: (/L—lf&)= 88 _-gl l--Z0 ~ TE I-lo _i[- Glo! 8 I/7 I/Z»`~g I/75 g I/K 8* + 8 L' E The clock will gain or lose t=N(T——-T0): iE-—O-@;.—=12X60 >< 108 Eg- . 295. F =2.3 > 0oT NPOT 364. I :2a is ten times larger than in normal burning. e 365. C -- zi-X-·A. Solution. The capacitance of the capacitor C = zag-IE , The rezsistance of the capacitor after hllgng it with electrolyte R = é-E-and its conductivity A = -F=—2- . Hence, C=-ZE);-A. The expression obtained is of a general nature, is true of capacitors gf any shape and is widely utilized in electrical engineering calculaions. 366. Lenz’s unit of current is 0.065 A.

CHAPTER III. ELECTRICITY 245 Solution. It follows from the laws of electrolysis that the quantity of substance liberated on one electrode M = é.- %I t where F is Faraday’s number; I is current in amperes; t is time in seconds and n is the gram—equivalent of the substance. On the basis of Avogadro’s law, the volume of oxygen evolved when the cu1·rent is passed will take 1/3 of the volume of detonating gas at a pressure of 760 mm Hg and will be equal to V = 13.72 cms. Hence, the mass of the liberated oxygen M = dV = 0.0196 g. The cu1·rent ,,,,MFn corresp0nd1ng to Lenz s umt will be I =7A367. R, = 10 ohms; R2 = 20 ohms; R3 = 60 ohms. Solution. When the rheostat is cut out the current is I 0 = 2; = = 4 A. The resistance Ri can be found from the equation 0 Ri+R0‘-Z—i·—— or Ri=—L—R0 Therefore, V Rz=·j*·§"·(R1+R0) and V Rs=·]···_j·(R2+R1+Ro) 0 368. The galvanometer should be cut into the circuit in series. The scale of the instrument will be: oo; 1.2 >< 107 ohms; 6 >< 10° ohms; 4.0 >< 10° ohms; ...; Egg >< 10° ohms where n is the number of the division. The minimum resistance that can be measured is 3 >< 105 ohms. Note. The values of the resistance Rn corresponding to separate divisions on the galvanometer scale can be determined from the formula V Rn::m where V is the mains voltage; n is the number of the scale division and I 0 is the current corresponding to one d1v1s1on on the galvanometer scale. 369. I0 = 5.05 A. _ _ _ _ _ Solution. If the voltage in the cu;/cuit IS V, the current in 1t before the amineter is cut in will be I 0 = F, and after the ammeter is cut .V 1n I = 1-- . Hence, 1 R+Ro I .._1i.”l'..1E ji 0** R -

246 ANSWERS AND SOLUTIONS R0 _ 370. R:.-;=50 ohms (n-20). Note. The response of the galvanometer can be reduced n. times if a current of $1 is passed through the shunt when the current in the circuit is I. 371. R = 0.032 ohm. The response of the instrument will be diminished 250 times. 372. 0.5 V per division. Note. For a current I = 1 mA to flow through the instrument its terminals should have a voltage of V=IR=10‘3 X 500= 0.5 V 373. Before the voltmeter was connected the voltage was V == = 105 V. The error is 5 V. 374. R = $5% == 61.2 ohms. The resistance R' calculated on the assumption that R0 -> oo will be 1.2 ohms less than the actual value. 375. For the 1 ohm resistance the measurement error using circuit a will be 0.1 ohm or 10 per cent, and using circuit b it will be 0.001 ohm or 0.1 per cent; for the 500 ohms resistance—0.1 ohm or 0.02 per cent and 167 ohms or 33.4 per cent, respectively. Solution. If V and I are the readings of the voltmeter and the ammeter, the resistance R' :5% calculated from these readings will be equal to the total resistance of the section in the circuit bb' when measurements are made using (a) and to the resistance of the section cc' when measuring by (b), i.e., it will he related to the resistance R by R' = R —{- R in the first case and by 1 G Rl __ HRD 2* R+R.. in the second. Comparing the values calculated from these ratios with the actual value of the resistance R one can find the errors permitted when measurements are made using (o) and (b). These errors are caused by the fact that in the first case (a) the voltage drop across the internal resistance of the ammeter is not subtracted from the voltmeter readings in calculations, and in the second case (b) the current taken by the voltmeter is not subtracted from the readings of the ammeter. Therefore, the resistance R' calculated only from the readings of the instruments is in the Hrst case larger and in the second case smaller than the actual resistance R. When the resistance R being measured drops in (a), the reduction of voltage shown by the ammeter will form an increasing share of the readings of the voltmeter and the circuit will produce increasingly larger relative errors. When the resistance is reduced in (b) the current taken by the voltmeter diminishes. The error in the readings of the

CHAPTER III. ELECTRICITY 247 ammeter and therefore the relative error in the calculation will also diminish. The circuit (b) is more effective for low resistances and (a) for higher ones. 376. V = Sl.2 V. Solution. The resistance of the section AB is R=.j>&... R0-|-2Ri The current flowing through the potentiometer is [-..3L. The voltage taken from the potentiometer is _ _ 2VRi _ 377. By 10.4 V. Solution. The resistances of the lamp and the appliance are 2 121-:-%%:240 ohms and R2=TI/@-:60 ohms The resistance of the circuit before and after the appliance is switched on is R'=R0-|-Ri:-.246 ohms and R"=R0-|— =54 ohms The current in the circuit before and after the appliance is switched on IS V B 12 I i=-I? Q 0.49 A, ` VE· IZZTZ A K The voltage drop in the wires is V6=IiR0=2.9 V and A v;=1212,,=1s.s v FIS- 296 I 2pl , 5 =L‘...• I \ l \\ { / g \ I \ 8 B/’ l \\ I Pi-- *2* { \\ \\ // ...."h_“."Q;. . ..u_, ,/ x I/// l ////// A // 11,1*/ gr Fig. 328 Fig. 329 after reflection from the mirror. The rays coming from O will be inside the bands AE and CF. The rays coming from all the points on the object will only arrive at each point in space between the straight lines 0’ ——-— —--F /4} I// 0 `iriél s//’ / ...**.... -.--- 5 [ 0 //// 0/X cz 6* Q4 z, 6 zz .0 F 0 Fig. 330 Fig. 331 AB and CF. The eye can only see the entire image of the object if it is at one of the points enclosed between the rays AB and CF. 515. By 10 cm. 517. At an angle of 45°. 518. on = 2°; as = l tance xs loc = 5 X 0.035 = 17.5 cm. 519. 3 m. _ _ Solution. The image of the wall will be behind the mirror at a distance lz = 4 m. If the eye is placed at the point A (Fig. 331) it will see only the rays coming from all the points in the section of

278 ANSWERS AND SOLUTIONS the wall image DE after the reflection in the mirror BC. Thus, the section of the wall visible in the mirro1· will have dimensions Z1 520. Only when the eye is placed inside the triangle DEH limited by the rays DG and EF (Fig. 332). F G CAH ”= ‘ ~·`$ { Ir "`°'T'“""""`T“"`;'>/ V yr v { /{/"‘ ; ,.·-•·T" // // `N/»’ E U, A, Fig. 332 521. The per endiculars to the mirrors A1 and A3 (see Fig. 173) should make angles of 22°30' with the incident rays, and the perpendiS /,’| C I E/ \`xL~\ I 1/ lu D fl /”/l •‘ \\ .,3 /,/’ "¤`ffIZ?#" ` / 9 S 3'éi/*"/I, // /,/’ I { ,/ I/’ » ”"‘{ 1 / ,/’ { ,/ \ \` a 4»’ 4/ xo 1.*,, 4 ...... .. .... A S Fig. 333 Fig. 334 culars to the mirrors A3 and A3—angles of 77°30'. The height of all the mirrors should not be less than the diameter of the lenses d, the Width of the mirrors A3 and A2 should b6 Gqual to i.O3d and the width of the mirrors A3 and A3 should be 2.6id.

CHAPTER IV. OPTICS 279 522. See Fig. 333; on = 120°. 523. The centre of the circle lies at the point O where the minors intersect. 524. Inside the band limited by the rays OC and OD (Fig. 334). 32. Spherical Mirrors 526. f= Vpq = 40 cm. 527. Solution. It follows from the similarity of the triangles ABF and CDF (Fig. 335) that G ig- = 1, ·· { .... 1......-.. B li ai-, _ 0 f \\ F Z' and from the concave mirror for- A mula 0 Lz a ··_· s H 2 ¤1-f Q2 • °" ,2 _ ,,2 Fig. sas TI “ `ZI 528. a2=—§- m; R=0.50 cm; f=25 cm (see Problem 527). 529. f= 2.5 cm. Note. If ai and az are the initial distances of the object and the image from the mirror, Z, and lg are the respective lengths of the object and the image, and ag is the distance from the image after the object is moved, the focal length can be found from the following system of equations: li ....2...4 ........“i°"b.-2 J. L...L d 1 1 -1 lz_¢z—° as _, a1+¤z—f an G1-b+¤s—f 53{)_ ..2 = L _ Z1 P I Note. The ratio -f- can be found from the equations: 1 l 7§··=·£?— . ¢11=P+f• a2'-= 2 the ray will be subjected to full internal reflection at the second face and emerge from the prism through the third face perpendicularly to the latter, 548. Tl··..·=.........._.....,.. =:¤1..3{t. Solution. If a bright image of the point source appears on the upper boundary of the plate, the rays coming from the source into the plate at small angles i (Fig. 346) pass unhin ered through the lower boundary. If the angle of incidence of the rays on the lower boundary is larger

CHAPTER rv. oprrcs 285 than the critical angle the rays are subjected to total internal reflection, illuminate the sensitive photolayer from below and form a halo. The refractive index can be calculated directly from Fig. 346 and from the definition of the critical angle ! Ta Sill i' =-L Iy~ _ ”· - -ml, =? T`———;._ --·‘% %_-;=_""Z;§s,- -5. QS [. .I I Sl Fig. 345 Fig. 346 34. Lenses and Composite Optical Systems 551. d = fq> = 2.35,mm. N ate. See Fig. 347. The image of the Sun will be in the focal plane of the lens and will be seen from the optical centre of the lens at an d 1.; ....’° r Fig. 347 angle q> just like the Sun. Since the angle qa is small it may be assumed that tan up z qa 552. In the plane passing through the optical centre of the_ lens. Note. To prove this it is enough to follow the motion of the image when the object is brought up to the lens (Fig. 3482; _ Remember that in any position of the object t e direction of the ray AF remains constant.,.The ray passing through O slowly_ turns about this point, as the object is brought up to the lens, making an increasing angle with the optical ams.

286 ANSWERS AND SOLUTIONS 553. See Fig. 349. 554. It will move 5 cm closer to the screen. N ate. In calculating the new position of the agex of the beam from the thin lens formula the apex of the beam A s ould be regarded as an object. l` `*`:Z:~~——·.- __ , I " \"`—— A r \}\ \\\ § i____-__.... -Q.;% 0 ._._.--._--. F .. §§ . sa Fig. 348 Remember that although in the normal case (the point of the object is the apex of the diverging beam of rays) the apex of the beam lies on the side where the rays are incident on the lens, in this particular case the apex of the beam lies on the other side of the lens with respect IAA a//z Q /*/// g Shy ,// // g § /’/ // g ·*I•—-••—·-O-·-——— g X - {1-——-———•-—--—--i Fig. 356 For construction it is convenient to take the rays going parallel to the optical axis and through the front focus of the lens. 568. The source should be at infinity. 569. See Fig. 357; ag = 2.5 cm. 570. If the distance between the halves is infinitely small, the beam will, for all intents and purposes, remain parallel. lf this distance l . —`·§ / . L ’F `, ’ F I 1 -_ 1I __ t i 1- _—— { Fig. 357 is large but less than the focal length of each half, the beam of parallel rays will be transformed into a beam of _converging rays. When the distance between the lenses 1S larger thanthe focal length of each, the parallelbeam, will be converte by the system into Ei diverging beam.

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